# Defining the Poisson distribution

The Poisson distribution is a family of discrete distributions with positive probabilities on the non-negative numbers $0,1,2,\cdots$. Each distribution in this family is indexed by a positive number $\lambda>0$. One way to define this distribution is to give its probability function given the parameter $\lambda$ and then derive various distributional quantities such as mean and variance. Along with other mathematical facts, it can be shown that both the mean and the variance are $\lambda$. In this post, we take a different tack. We look at two view points that give rise to the Poisson distribution. Taking this approach will make it easier to appreciate some of the possible applications of the Poisson distribution. The first view point is that the Poisson distribution is the limiting case of the binomial distribution. The second view point is through the Poisson process, a stochastic process that, under some conditions, counts the number of events and the time points at which these events occur in a given time (or physical) interval.

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Poisson as a limiting case of binomial

A binomial distribution where the number of trials $n$ is large and the probability of success $p$ is small such that $np$ is moderate in size can be approximated using the Poisson distribution with mean $\lambda=np$. This fact follows from Theorem 1, which indicates that the Poisson distribution is the limiting case of the binomial distribution.

Theorem 1
Let $\lambda$ be a fixed positive constant. Then for each integer $x=0,1,2,\cdots$, the following is true:

$\displaystyle \lim_{n \rightarrow \infty} \binom{n}{x} \ p^x \ (1-p)^{n-x}=\lim_{n \rightarrow \infty} \frac{n!}{x! \ (n-x)!} \ p^x \ (1-p)^{n-x}=\frac{e^{-\lambda} \ \lambda^x}{x!}$

where $\displaystyle p=\frac{\lambda}{n}$.

Proof of Theorem 1
We start with a binomial distribution with $n$ trials and with $\displaystyle p=\frac{\lambda}{n}$ being the probability of success, where $n>\lambda$. Let $X_n$ be the count of the number of successes in these $n$ Bernoulli trials. The following is the probability that $X_n=k$.

\displaystyle \begin{aligned} P(X_n=k)&=\binom{n}{k} \biggl(\frac{\lambda}{n}\biggr)^k \biggr(1-\frac{\lambda}{n}\biggr)^{n-k} \\&=\frac{n!}{k! (n-k)!} \biggl(\frac{\lambda}{n}\biggr)^k \biggr(1-\frac{\lambda}{n}\biggr)^{n-k} \\&=\frac{n(n-1)(n-2) \cdots (n-k+1)}{n^k} \biggl(\frac{\lambda^k}{k!}\biggr) \biggr(1-\frac{\lambda}{n}\biggr)^{n} \biggr(1-\frac{\lambda}{n}\biggr)^{-k} \\&=\biggl(\frac{\lambda^k}{k!}\biggr) \ \biggl[ \frac{n(n-1)(n-2) \cdots (n-k+1)}{n^k} \ \biggr(1-\frac{\lambda}{n}\biggr)^{n} \ \biggr(1-\frac{\lambda}{n}\biggr)^{-k} \biggr] \end{aligned}

In the last step, the terms that contain $n$ are inside the square brackets. Let’s see what they are when $n$ approaches infinity.

$\displaystyle \lim \limits_{n \rightarrow \infty} \ \frac{n(n-1)(n-2) \cdots (n-k+1)}{n^k}=1$

$\displaystyle \lim \limits_{n \rightarrow \infty} \biggr(1-\frac{\lambda}{n}\biggr)^{n}=e^{-\lambda}$

$\displaystyle \lim \limits_{n \rightarrow \infty} \biggr(1-\frac{\lambda}{n}\biggr)^{-k}=1$

The reason that the first result is true is that the numerator is a polynomial where the leading term is $n^k$. Upon dividing by $n^k$ and taking the limit, we get 1. The second result is true since the following limit is one of the definitions of the exponential function $e^x$.

$\displaystyle \lim \limits_{n \rightarrow \infty} \biggr(1+\frac{x}{n}\biggr)^{n}=e^{x}$

The third result is true since the exponent $-k$ is a constant. Thus the following is the limit of the probability $P(X_n=k)$ as $n \rightarrow \infty$.

\displaystyle \begin{aligned} \lim \limits_{n \rightarrow \infty} P(X_n=k)&= \biggl(\frac{\lambda^k}{k!}\biggr) \ \lim \limits_{n \rightarrow \infty} \biggl[ \frac{n(n-1)(n-2) \cdots (n-k+1)}{n^k} \ \biggr(1-\frac{\lambda}{n}\biggr)^{n} \ \biggr(1-\frac{\lambda}{n}\biggr)^{-k} \biggr] \\&=\biggl(\frac{\lambda^k}{k!}\biggr) \cdot 1 \cdot e^{-\lambda} \cdot 1 \\&=\frac{e^{-\lambda} \lambda^k}{k!} \end{aligned}

This above derivation completes the proof. $\blacksquare$

In a given binomial distribution, whenever the number of trials $n$ is large and the probability $p$ of success in each trial is small (i.e. each of the Bernoulli trial rarely results in a success), Theorem 1 tells us that we can use the Poisson distribution with parameter $\lambda=np$ to estimate the binomial distribution.

Example 1
The probability of being dealt a full house in a hand of poker is approximately 0.001441. Out of 5000 hands of poker that are dealt at a certain casino, what is the probability that there will be at most 4 full houses?

Let $X$ be the number of full houses in these 5000 poker hands. The exact distribution for $X$ is the binomial distribution with $n=$ 5000 and $p=$ 0.001441. Thus example deals with a large number of trials where each trial is a rare event. So the Poisson estimation is applicable. Let $\lambda=$ 5000(0.001441) = 7.205. Then $P(X \le 4)$ can be approximated by the Poisson random variable $Y$ with parameter $\lambda$. The following is the probability function of $Y$:

$\displaystyle P(Y=y)=e^{-7.205} \ \frac{7.205^y}{y!}$

The following is the approximation of $P(X \le 4)$:

\displaystyle \begin{aligned} P(X \le 4)&\approx P(Y \le 4) \\&=P(Y=0)+P(Y=1)+P(Y=2)+P(Y=3)+P(Y=4) \\&= e^{-7.205} \biggl[ 1+7.205+\frac{7.205^2}{2!}+\frac{7.205^3}{3!}+\frac{7.205^4}{4!}\biggr] \\&=0.155098087 \end{aligned}

The following is a side by side comparison between the binomial distribution and its Poisson approximation. For all practical purposes, they are indistingusihable from one another.

$\displaystyle \begin{bmatrix} \text{Count of}&\text{ }&\text{ }&\text{Binomial } &\text{ }&\text{ }&\text{Poisson } \\\text{Full Houses}&\text{ }&\text{ }&P(X \le x) &\text{ }&\text{ }&P(Y \le x) \\\text{ }&\text{ }&\text{ }&n=5000 &\text{ }&\text{ }&\lambda=7.205 \\\text{ }&\text{ }&\text{ }&p=0.001441 &\text{ }&\text{ }&\text{ } \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 0&\text{ }&\text{ }&0.000739012&\text{ }&\text{ }&0.000742862 \\ 1&\text{ }&\text{ }&0.006071278&\text{ }&\text{ }&0.006095184 \\ 2&\text{ }&\text{ }&0.025304641&\text{ }&\text{ }&0.025376925 \\ 3&\text{ }&\text{ }&0.071544923&\text{ }&\text{ }&0.071685238 \\ 4&\text{ }&\text{ }&0.154905379&\text{ }&\text{ }&0.155098087 \\ 5&\text{ }&\text{ }&0.275104906&\text{ }&\text{ }&0.275296003 \\ 6&\text{ }&\text{ }&0.419508250&\text{ }&\text{ }&0.419633667 \\ 7&\text{ }&\text{ }&0.568176421 &\text{ }&\text{ }&0.568198363 \\ 8&\text{ }&\text{ }&0.702076190 &\text{ }&\text{ }&0.701999442 \\ 9&\text{ }&\text{ }&0.809253326&\text{ }&\text{ }&0.809114639 \\ 10&\text{ }&\text{ }&0.886446690&\text{ }&\text{ }&0.886291139 \\ 11&\text{ }&\text{ }&0.936980038&\text{ }&\text{ }&0.936841746 \\ 12&\text{ }&\text{ }&0.967298041&\text{ }&\text{ }&0.967193173 \\ 13&\text{ }&\text{ }&0.984085073&\text{ }&\text{ }&0.984014868 \\ 14&\text{ }&\text{ }&0.992714372&\text{ }&\text{ }&0.992672033 \\ 15&\text{ }&\text{ }&0.996853671&\text{ }&\text{ }&0.996830358 \end{bmatrix}$

The above table is calculated using the functions BINOM.DIST and POISSON.DIST in Excel. The following shows how it is done. The parameter TRUE indicates that the result is a cumulative distribution. When it is set to FALSE, the formula gives the probability function.

$P(X \le x)=\text{BINOM.DIST(x, 5000, 0.001441, TRUE)}$

$P(Y \le x)=\text{POISSON.DIST(x, 7.205, TRUE)}$

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The Poisson distribution

The limit in Theorem 1 is a probability function and the resulting distribution is called the Poisson distribution. We now gives the formal definition. A random variable $X$ that takes on one of the numbers $0,1,2,\cdots$ is said to be a Poisson random variable with parameter $\lambda>0$ if

$\displaystyle P(X=x)=\frac{e^{-\lambda} \ \lambda^x}{x!} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=0,1,2,\cdots$

It can be shown that the above function is indeed a probability function, i.e., the probabilities sum to 1. Any random variable that has a probability function of the above form is said to follow (or to have) a Poisson distribution. Furthermore, it can be shown that $E(X)=var(X)=\lambda$, i.e., the Poisson parameter is both the mean and variance. Thus the Poisson distribution may be a good fit if the observed data indicate that the sample mean and the sample variance are nearly identical.

The following is the moment generating function of the Poisson distribution with parameter $\lambda$.

$\displaystyle M(t)=E(e^{tX})=e^{\lambda \ (e^t-1)}$

One consequence of the Poisson moment generating function is that any independent sum of Poisson distributions is again a Poisson distribution.

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The Poisson process

Another way, the more important way, to look at the Poisson distribution is the view point of the Poisson process. Consider an experiment in which events that are of interest occur at random in a time interval. The goal here is to record the time of the occurrence of each random event and for the purpose at hand, count the number of random events occurring in a fixed time interval. Starting at time 0, note the time of the occurrence of the first event. Then the time at which the second random event occurs and so on. Out of these measurements, we can derive the length of time between the occurrences of any two consecutive random events. Such measurements belong to a continuous random variable. In this post, we focus on the discrete random variable of the count of the random events in a fixed time interval.

A good example of a Poisson process is the well known experiment in radioactivity conducted by Rutherford and Geiger in 1910. In this experiment, $\alpha$-particles were emitted from a polonium source and the number of $\alpha$-particles were counted during an interval of 7.5 seconds (2608 many such time intervals were observed). A Poisson process is a random process in which several criteria are satisfied. We will show that in a Poisson process, the number of these random occurrences in the fixed time interval will follow a Poisson distribution. First, we discuss the criteria to which a Poisson process must conform.

One of the criteria is that in a very short time interval, the chance of having more than one random event is essentially zero. So either one random event will occur or none will occur in a very short time interval. Considering the occurrence of a random event as a success, there is either a success or a failure in a very short time interval. So a very short time interval in a Poisson process can be regarded as a Bernoulli trial.

The second criterion is that the experiment remains constant over time. Specifically this means that the probability of a random event occurring in a given subinterval is proportional to the length of that subinterval and not on where the subinterval is in the original interval. For example, in the 1910 radioactivity study, $\alpha$-particles were emitted at the rate of $\lambda=$ 3.87 per 7.5 seconds. So the probability of one $\alpha$-particle emitted from the radioactive source in a one-second interval is 3.87/7.5 = 0.516. Then the probability of observing one $\alpha$-particle in a half-second interval is 0.516/2 = 0.258. For a quarter-second interval, the probability is 0.258/2 = 0.129. So if we observe half as long, it will be half as likely to observe the occurrence of a random event. On the other hand, it does not matter when the quarter-second subinterval is, whether at the beginning or toward the end of the original interval of 7.5 seconds.

The third criterion is that non-overlapping subintervals are mutually independent in the sense that what happens in one subinterval (i.e. the occurrence or non-occurrence of a random event) will have no influence on the occurrence of a random event in another subinterval. To summarize, the following are the three criteria of a Poisson process:

Suppose that on average $\lambda$ random events occur in a time interval of length 1.

1. The probability of having more than one random event occurring in a very short time interval is essentially zero.
2. For a very short subinterval of length $\frac{1}{n}$ where $n$ is a sufficiently large integer, the probability of a random event occurring in this subinterval is $\frac{\lambda}{n}$.
3. The numbers of random events occurring in non-overlapping time intervals are independent.

Consider a Poisson process in which the average rate is $\lambda$ random events per unit time interval. Let $Y$ be the number of random events occurring in the unit time interval. In the 1910 radioactivity study, the unit time interval is 7.5 seconds and $Y$ is the count of the number of $\alpha$-particles emitted in 7.5 seconds. It follows that $Y$ has a Poisson distribution with parameter $\lambda$. To see this, subdivide the unit interval into $n$ non-overlapping subintervals of equal length where $n$ is a sufficiently large integer. Let $X_{n,j}$ be the number of random events in the the $j$th subinterval ($1 \le j \le n$). Based on the three assumptions, $X_{n,1},X_{n,2},\cdots,X_{n,n}$ are independent Bernoulli random variables, where the probability of success for each $X_{n,j}$ is $\frac{\lambda}{n}$. Then $X_n=X_{n,1}+X_{n,2}+\cdots+X_{n,n}$ has a binomial distribution with parameters $n$ and $p=\frac{\lambda}{n}$. Theorem 1 tells us that the limiting case of the binomial distributions for $X_n$ is the Poisson distribution with parameter $\lambda$. This Poisson distribution should agree with the distribution for $Y$. The Poisson is also discussed in quite a lot of details in the previous post called Poisson as a Limiting Case of Binomial Distribution.

We now examine the 1910 radioactivity study a little more closely.

Example 2
The basic idea of the 1910 radioactivity study conducted by Rutherford and Geiger is that a polonium source was placed a short distance from an observation point. The number of $\alpha$-particles emitted from the source were counted in 7.5-second intervals for 2608 times. The following is the tabulated results.

$\displaystyle \begin{bmatrix} \text{Number of alpha particles}&\text{ }&\text{Observed} \\ \text{recorded per 7.5 seconds }&\text{ }&\text{counts} \\ \text{ }&\text{ }&\text{ } \\ 0&\text{ }&57 \\ 1&\text{ }&203 \\ 2&\text{ }&383 \\ 3&\text{ }&525 \\ 4&\text{ }&532 \\ 5&\text{ }&408 \\ 6&\text{ }&273 \\ 7&\text{ }&139 \\ 8&\text{ }&45 \\ 9&\text{ }&27 \\ 10&\text{ }&10 \\ 11+&\text{ }&6 \\ \text{ }&\text{ }&\text{ } \\ \text{Total }&\text{ }&2608 \end{bmatrix}$

What is the average number of particles observed per 7.5 seconds? The total number of $\alpha$-particles in these 2608 periods is

$0 \times 57+1 \times 203+2 \times 383+ 3 \times 525 + \cdots=10097$.

The mean count per period is $\lambda=\frac{10097}{2608}=3.87$. Consider the Poisson distribution with parameter 3.87. The following is its probability function.

$\displaystyle P(X=x)=\frac{e^{-3.87} \ 3.87^x}{x!} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=0,1,2,\cdots$

Out of 2608 periods, the expected number of periods with $x$ particles in emission is $2608P(X=x)$. The following is a side by side comparison in the observed counts and the expected counts.

$\displaystyle \begin{bmatrix} \text{Number of alpha particles}&\text{ }&\text{Observed}&\text{ }&\text{Expected} \\ \text{recorded per 7.5 seconds }&\text{ }&\text{counts}&\text{ }&\text{counts} \\ \text{ }&\text{ }&\text{ }&\text{ }&2608 \times P(X=x) \\ \text{ }&\text{ }&\text{ }&\text{ }&\text{ } \\ 0&\text{ }&57&\text{ }&54.40 \\ 1&\text{ }&203&\text{ }&210.52 \\ 2&\text{ }&383&\text{ }&407.36 \\ 3&\text{ }&525&\text{ }&525.50 \\ 4&\text{ }&532&\text{ }&508.42 \\ 5&\text{ }&408&\text{ }&393.52 \\ 6&\text{ }&273&\text{ }&253.82 \\ 7&\text{ }&139&\text{ }&140.32 \\ 8&\text{ }&45&\text{ }&67.88 \\ 9&\text{ }&27&\text{ }&29.19 \\ 10&\text{ }&10&\text{ }&11.30 \\ 11+&\text{ }&6&\text{ }&5.78 \\ \text{ }&\text{ }&\text{ }&\text{ }&\text{ } \\ \text{Total }&\text{ }&2608&\text{ }&2608 \end{bmatrix}$

The expected counts are quite close to the observed counts, showing that the Poisson distribution is a very good fit to the observed data from the 1910 study.

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We have described the Poisson process as the distribution of random events in a time interval. The same idea can be used to describe random events occurring along a spatial interval, i.e. intervals in terms of distance or volume or other spatial measurements (see Examples 5 and 6 below).

Another point to make is that sometimes it may be necessary to consider an interval other than the unit length. Instead of counting the random events occurring in an interval of length 1, we may want to count the random events in an interval of length $t$. As before, let $\lambda$ be the rate of occurrences in a unit interval. Then the rate of occurrences of the random events is over the interval of length $t$ is $\lambda t$. The same idea will derive that fact that the number of occurrences of the random events of interest in the interval of length $t$ is a Poisson distribution with parameter $\lambda t$. The following is its probability function.

$\displaystyle P(X_t=x)=\frac{e^{-\lambda t} \ (\lambda t)^x}{x!} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=0,1,2,\cdots$

where $X_t$ is the count of the random events in an interval of length $t$.

For example, in the 1910 radioactive study, the unit length is 7.5 seconds. The number of $\alpha$-particles observed in a half unit interval (3.75 seconds) will follow a Poisson distribution with parameter $0.5 \lambda=$ 0.5(3.87) = 1.935 with the following probability function:

$\displaystyle P(X_{0.5}=x)=\frac{e^{-1.935} \ (1.935)^x}{x!} \ \ \ \ \ \ \ \ \ \ \ \ \ x=0,1,2,\cdots$

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More examples

Example 3
A radioactive source is metered for 5 hours. During this period, 9638 $\alpha$-particles are counted. What is the probability that during the next minute, between 30 and 34 particles (both inclusive ) will be counted?

The average number of $\alpha$-particles counted per minute is $\lambda=\frac{9638}{300}=32.12$. Let $X$ be the number of $\alpha$-particles counted per minute. Then $X$ has a Poisson distribution with parameter $\lambda=32.12$. The following calculates $P(30 \le X \le 34)$.

\displaystyle \begin{aligned} P(30 \le X \le 34)&=e^{-32.12} \biggl[ \frac{32.12^{30}}{30!}+\frac{32.12^{31}}{31!}+\frac{32.12^{32}}{32!}+\frac{32.12^{33}}{33!}+\frac{32.12^{34}}{34!} \biggr] \\&=0.341118569 \end{aligned}

Alternatively, the POISSON.DIST function in Excel can be used as follows:

\displaystyle \begin{aligned} P(30 \le X \le 34)&=P(X \le 34)-P(X \le 29) \\&=\text{POISSON.DIST(34,32.12,TRUE)} \\& \ \ \ \ \ \ -\text{POISSON.DIST(29,32.12,TRUE)} \\&=0.671501917-0.330383348 \\&=0.341118569 \end{aligned}

Example 4
The side effect of dry mouth is known to be experienced, on the average, by 5 out of 10,000 individuals taking a certain medication. About 20,000 patients are expected to take this medication next year. What is the probability that between 12 and 16 (both inclusive) patients will experience the side effect of dry mouth? What is the exact probability model that can also be used to work this problem?

The exact model is a binomial distribution. The number of trials $n=$ 20000 and the probability of success in each trial is $p=$ 0.0005 (experiencing the side effect). Here, we use Poisson to estimate the binomial. The average number of patients experiencing side effect is $\lambda=20000(0.0005)=10$. Let $X$ be the number of patients experiencing the side effect. The following calculates the Poisson probability for $P(12 \le X \le 16)$ in two different ways.

\displaystyle \begin{aligned} P(12 \le X \le 16)&=e^{-10} \biggl[ \frac{10^{12}}{12!}+\frac{10^{13}}{13!}+\frac{10^{14}}{14!}+\frac{10^{15}}{15!}+\frac{10^{16}}{16!} \biggr] \\&=0.276182244 \end{aligned}
\displaystyle \begin{aligned} P(12 \le X \le 16)&=P(X \le 11)-P(X \le 16) \\&=\text{POISSON.DIST(16,10,TRUE)} \\& \ \ \ \ \ \ -\text{POISSON.DIST(11,10,TRUE)} \\&=0.97295839-0.696776146 \\&=0.276182244 \end{aligned}

Example 5
In a 10-mile stretch of a highway, car troubles (e.g. tire punctures, dead batteries, and mechanical breakdown) occur at a rate of 1.5 per hour. A tow truck driver can respond to such car troubles and offer roadside assistance, which can include towing and minor repair. Assume that the number of such incidences per hour follows a Poisson distribution. At the beginning of the hour, three tow trucks (and their drivers) are available to respond to any car troubles in this stretch of highway. What is the probability that in the next hour all three tow trick drivers will be busy helping motorists with car troubles in this stretch of highway?

Let $X$ be the number of car troubles that occur in this 10-mile stretch of highway in the one-hour period in question. If in this one hour there are 3 or more car troubles ($X \ge 3$), then all three tow truck drivers will be busy.

\displaystyle \begin{aligned} P(X \ge 3)&=1-P(X \le 2) \\&=1-e^{-1.5} \biggl[ 1+1.5+\frac{1.5^{2}}{2!} \biggr] \\&=1-0.808846831\\&=0.191153169 \end{aligned}

Example 6
Continuing Example 5. Considering that there is only 19% chance that all 3 tow truck drivers will be busy, there is a good chance that the resources are under utilized. What if one of the drivers is assigned to another stretch of highway?

With only two tow trucks available for this 10-mile stretch of highway, the following is the probability that all two tow truck drivers will be busy:

\displaystyle \begin{aligned} P(X \ge 2)&=1-P(X \le 1) \\&=1-e^{-1.5} \biggl[ 1+1.5 \biggr] \\&=1-0.5578254\\&=0.4421746 \end{aligned}

Assigning one driver to another area seems to be a better way to make good use of the available resources. With only two tow truck drivers available, there is much reduced chance (56%) that one of the drivers will be idle, and there is a much increased chance (44%) that all available drivers will be busy.

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Remarks

The Poisson distribution is one of the most important of all probability models and has shown to be an excellent model for a wide array of phenomena such as

• the number of $\alpha$-particles emitted from radioactive source in a given amount of time,
• the number of vehicles passing a particular location on a busy highway,
• the number of traffic accidents in a stretch of highway in a given period of time,
• the number of phone calls arriving at a particular point in a telephone network in a fixed time period,
• the number of insurance losses/claims in a given period of time,
• the number of customers arriving at a ticket window,
• the number of earthquakes occurring in a fixed period of time,
• the number of mutations on a strand of DNA.
• the number of hurricanes in a year that originate in the Atlantic ocean.

What is the Poisson distribution so widely applicable in these and many other seemingly different and diverse phenomena? What is the commonality that ties all these different and diverse phenomena? The commonality is that all these phenomena are basically a series of independent Bernoulli trials. If a phenomenon is a Binomial model where $n$ is large and $p$ is small, then it has a strong connection to Poisson model mathematically through Theorem 1 above (i.e. it has a Poisson approximation). On the other hand, if the random phenomenon follows the criteria in a Poisson process, then the phenomenon is also approximately a Binomial model, which means that in the limiting case it is Poisson.

In both view points discussed in this post, the Poisson distribution can be regarded as a binomial distribution taken at a very granular level. This connection with the binomial distribution points to a vast arrays of problems that can be solved using the Poisson distribution.

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Exercises

Practice problems for the Poisson concepts discussed above can be found in the companion blog (go there via the following link). Working on these exercises is strongly encouraged (you don’t know it until you can do it).

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$\copyright \ \text{2015 by Dan Ma}$

# A natural look at the negative binomial survival function

The negative binomial distribution is a discrete distribution with two parameters $r$ and $p$ where $r>0$ and $0. It has positive probabilities at the non-negative integers $0,1,2,\cdots$. So it can potentially be used as a model for the random count of a phenomenon of interest. In some cases, the negative binomial distribution has a natural interpretation. In fact, the natural interpretation should be how the negative binomial distribution is introduced. With the parameter $r$ being a positive integer, the negative binomial distribution can be naturally interpreted as the discrete waiting time until the $r$th success. But this natural interpretation does not apply to the general case of $r$ being only a positive real number but not necessarily an integer. However, once the natural case that $r$ being a positive integer is understood, it is easy to make a leap to the general case that $r$ is any positive real number. In this post, we focus on the “natural” version of the negative binomial distribution, where $r$ is a positive integer. In this case, there is a natural way to look at the cumulative distribution function (cdf) and the survival function. The discussion in this post will complement the following posts on the negative binomial distribution.

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The probability functions

A Bernoulli trial is a random experiment in which there are two distinct outcomes. For convenience, they are called success (S) and failure (F). Consider performing a sequence of independent Bernoulli trials such that each trial has the same probability of success $p$. Fix a positive integer $r$. In some cases, we would like to count the number of trials that produce the $r$th success. Let’s call this count $X_r$. In other cases, we may want instead to count the number of failures before getting the $r$th success. Let’s call this count $Y_r$. According to the discussion in the two posts listed above, the probability functions of $X_r$ and $Y_r$ are:

$\displaystyle P(X_r=x)= \binom{x-1}{r-1} p^r (1-p)^{x-r} \ \ \ \ \ \ \ x=r,r+1,r+2,\cdots \ \ \ \ \ \ \ (1)$

$\displaystyle P(Y_r=y)=\binom{y+r-1}{y} p^r (1-p)^y \ \ \ \ \ \ \ y=0,1,2,\cdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$

When $r=1$, the resulting distribution is called the geometric distribution.

$\displaystyle P(X_1=x)= p (1-p)^{x-1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=1,2,3,\cdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1a)$

$\displaystyle P(Y_1=y)= p (1-p)^y \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y=0,1,2,\cdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2a)$

Because the parameter $r$ is a positive integer, all the above probability functions have an combinatorial explanation. For the probability function in (1), there are $x$ many trials. The last Bernoulli trial is the $r$th success. Then all the preceding $x-1$ trails have at most $r-1$ successes. Hence we have the binomial coefficient $\binom{x-1}{r-1}$ in (1). For the probability function in (2), there are $y+r$ trials ($r$ successes and $y$ failures). Again, the last Bernoulli trial is the $r$th success. So in the preceding $y+r-1$ trials, there are exactly $r-1$ successes and exactly $y$ failures. Hence we have the binomial coefficient $\binom{y+r-1}{y}$ in (2). Let’s look at some examples.

Example 1
Four different prizes are randomly put into boxes of cereal. One of the prizes is a free ticket to the local zoo. Suppose that a family of four (called Family A) decides to buy this cereal until obtaining four free tickets to the zoo. What is the probability that the family will have to buy 10 boxes of cereal to obtain the four free tickets to the zoo? What is the probability that the family will have to buy 16 boxes of cereal to obtain the four free tickets to the zoo?

In this example, the success is a box of cereal with a free ticket to the zoo. So getting a ticket to the zoo is considered a success. Any one of the other three prizes is considered undesirable or a failure. Any one of the four prizes is equally likely. The negative binomial distribution in this example has $r=4$ and $p=0.25$. The count of the boxes of cereal to be purchased is the random variable $X_4$ as described in (1) above. The following gives the answers.

\displaystyle \begin{aligned} P(X_4=10)&=\binom{10-1}{3} \ (0.25)^4 \ (0.75)^{10-4} \\&=\binom{9}{3} \ (0.25)^4 \ (0.75)^{6} \\&=84 \ (0.25)^4 \ (0.75)^{6} \\&=0.0583992 \end{aligned}

\displaystyle \begin{aligned} P(X_4=16)&=\binom{16-1}{3} \ (0.25)^4 \ (0.75)^{16-4} \\&=\binom{15}{3} \ (0.25)^4 \ (0.75)^{12} \\&=455 \ (0.25)^4 \ (0.75)^{12} \\&=0.056299766 \end{aligned}

Example 2 (Example 1 continued)
Suppose Family A agrees to give any one of the undesirable prizes away to another family (called Family B). What is the probability that Family A will give 5 undesirable prizes to Family B before obtaining the four desirable tickets to the zoo? What is the probability that Family A will give 12 undesirable prizes to Family B before obtaining the four tickets to the zoo?

The negative binomial distribution in this example has $r=4$ and $p=0.25$. The interest here is to count the number failures (undesirable prizes) before getting 4 successes. Thus the random variable of interest is $Y_4$ as described in (2) above. The following gives the answers.

\displaystyle \begin{aligned} P(Y_4=5)&=\binom{5+4-1}{5} \ (0.25)^4 \ (0.75)^{5} \\&=\binom{8}{5} \ (0.25)^4 \ (0.75)^{5} \\&=56 \ (0.25)^4 \ (0.75)^{5} \\&=0.0519104 \end{aligned}

\displaystyle \begin{aligned} P(Y_4=12)&=\binom{12+4-1}{10} \ (0.25)^4 \ (0.75)^{12} \\&=\binom{15}{12} \ (0.25)^4 \ (0.75)^{12} \\&=455 \ (0.25)^4 \ (0.75)^{12} \\&=0.056299766 \end{aligned}

Here’s the mean and variance for both examples.

$\displaystyle E(X_4)=4 \ \frac{1}{0.25}=16$

$\displaystyle Var(X_4)=Var(Y_4)=4 \ \frac{0.75}{0.25^2}=48$

$\displaystyle E(Y_4)=4 \ \frac{0.75}{0.25}=12$

Thus Family A is expected to buy 16 boxes of cereal to get the 4 tickets to the zoo and is expected to give 12 prizes to the other family. However, the variance is fairly large. As a result, the actual number of boxes purchased can vary from the mean by a large amount.

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The survival functions and the cumulative distribution functions

For any random variable $T$, the cumulative distribution function is $P(T \le t)$ where $T$ can range over all real numbers $t$ or a relevant set of real numbers $t$. The survival function is $P(T > t)$. The term survival comes from the interpretation that $P(T > t)$ is the probability of a life surviving beyond time $t$ if $T$ is meant to model the lifetime of a biological life or some system. Even when $T$ is not a lifetime random variable, we still use the term survival function for $P(T > t)$.

Example 1 asks the probability of a certain number of trials in order to get $r$th success and the probability of a certain number of failures before getting the $r$th success. Sometimes it is more informative to know how many trials that are required to be performed in order to achieve one’s goal. For example, it may be useful to know the mean number of trials or the probability of achieving the goal in $x$ trials or less. In some cases, it may take time and resource to perform the random Bernoulli trials. It will be helpful to know ahead of time the likelihood of achieving one’s goal given the resources that are available. In the above examples, it will be helpful for Family A to have a better and clearer sense of how many boxes of cereal are to be purchased. Therefore, it is worthwhile to look at the cdf and the survival function of the negative binomial distribution.

In terms of basic principle, here’s the survival functions for the distribution described in (1) and (2).

\displaystyle \begin{aligned} P(X_r>x)&=\sum \limits_{k=x+1}^\infty P(X_r=k) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=r,r+1,r+2,\cdots \\&=\sum \limits_{k=x+1}^\infty \binom{k-1}{r-1} p^r (1-p)^{k-r} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3) \end{aligned}

\displaystyle \begin{aligned} P(Y_r>y)&=\sum \limits_{j=y+1}^\infty P(Y_r=j) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y= 0,1,2,3,\cdots \\&=\sum \limits_{j=y+1}^\infty \binom{j+r-1}{j} p^r (1-p)^{j} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4) \end{aligned}

The cumulative distribution functions can be defined by basic principle or by taking the complement as follows:

\displaystyle \begin{aligned} P(X_r \le x)&=\sum \limits_{k=r}^x P(X_r=k) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=r,r+1,r+2,\cdots \\&=1-P(X_r>x) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)\end{aligned}

\displaystyle \begin{aligned} P(Y_r \le y)&=\sum \limits_{j=0}^y P(Y_r=j) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y=0,1,2,3,\cdots \\&=1-P(Y_r>y) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)\end{aligned}

Example 3 (the above examples continued)
What is the probability that Family A will buy more than 16 boxes in order to obtain the 4 free tickets to the zoo? What is the probability that Family A will give away at most 10 undesirable prizes before obtaining the 4 free tickets to the zoo?

Working from the definition, here’s the answers:

$\displaystyle P(X_4>16)=1-P(X_4 \le 16)=0.40498711$

$\displaystyle P(Y_4 \le 10)=0.47866004$

The mean number of boxes of cereal purchased is 16 as indicated earlier. Since the variance is large, there is still a significance chance (about 40%) that Family A will have to buy more than 16 boxes of cereal before achieving their goal. On the other hand, there is good chance (about 48%) that Family will give away at most 10 undesirable prizes.

Note that the calculation for $P(X_4>16)$ is based on $P(X_4 \le 16)$, which requires the calculation of 17 probabilities. The calculation for $P(Y_4 \le 10)$, which requires 11 probabilities. Such calculation can be done by software of course. There is a natural way of looking at the calculation for (3), (4), (5) and (6). This alternative approach will give much better insight on the negative binomial distribution.

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A more natural way of interpreting the survival function

We now discuss a better way to look at the survival functions in (3) and (4). Consider the event $X_r>x$ and the event $Y_r>y$. We will see that the negative binomial survival function can be related to the cdf of a binomial distribution.

For the event $X_r>x$ to occur, the $r$th success occurs after performing $x$ trials. So it will take $x+1$ trials or more to get the $r$th success. This means that in the first $x$ trials, there are at most $r-1$ successes. The following highlights the equivalent statements.

\displaystyle \begin{aligned} X_r>x&\equiv \text{the } r \text{th success occurs after performing } x \text{ trials} \\&\equiv \text{it takes at least } x+1 \text{ trials to get the } r \text{th success} \\&\equiv \text{in the first } x \text{ trials, there are at most } r-1 \text{ successes} \end{aligned}

The last statement is a binomial distribution. Specifically, it is the binomial distribution with $x$ trials and the probability of success $p$. Let’s denote the count of successes of this binomial distribution by $B_{x,p}$. Thus we can relate the survival function in (3) with the cdf of $B_{x,p}$ as follows:

\displaystyle \begin{aligned} P(X_r>x)&=P(B_{x,p} \le r-1) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=r,r+1,r+2,\cdots \\&=\sum \limits_{k=0}^{r-1} \ P(B_{x,p}=k) \\&=\sum \limits_{k=0}^{r-1} \ \binom{x}{k} \ p^k (1-p)^{x-k} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (7) \end{aligned}

The advantage of (7) is that it gives us the insight of relating the negative binomial distribution with the binomial distribution. In terms of computation, (7) only requires $r$ many binomial probabilities. Thus Example 3 only requires the computation of 4 probabilities (versus 17 previously).

Note that $X_r=Y_r+r$. Thus the event $Y_r>y$ is the same as the event $X_r>y+r$. So we can just piggy back on the work done in (7). For the sake of the more clarity, here’s a translation for the event $Y_r>y$.

\displaystyle \begin{aligned} Y_r>y&\equiv X_r>y+r \\&\equiv \text{the } r \text{th success occurs after performing } y+r \text{ trials} \\&\equiv \text{it takes at least } y+r+1 \text{ trials to get the } r \text{th success} \\&\equiv \text{in the first } y+r \text{ trials, there are at most } r-1 \text{ successes} \end{aligned}

As before, let $B_{n,p}$ denote the number of successes in performing $n$ Bernoulli trials with $p$ as the probability of success. Based on the above translation, the following gives the survival function for the negative binomial random variable $Y_r$.

\displaystyle \begin{aligned} P(Y_r>y)&=P(X_r>y+r) \\&=P(B_{y+r,p} \le r-1) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y=0,1,2,\cdots \\&=\sum \limits_{j=0}^{r-1} \ P(B_{y+r,p}=j) \\&=\sum \limits_{j=0}^{r-1} \ \binom{y+r}{j} \ p^j (1-p)^{y+r-j} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (8) \end{aligned}

For both negative binomial random variables $X_r$ and $Y_r$, the survival functions can be computed using (7) and (8), respectively. Rote memorization of the formulas (7) and (8) is not recommended. Instead, focus on the thought process that translate the events $X_r>x$ and $Y_r>y$ into a binomial distribution.

Example 4 (the above examples continued)
We now rework Example 3 using the ideas presented in (7) and (8).

\displaystyle \begin{aligned} P(X_4>16)&=\sum \limits_{k=0}^{3} \ \binom{16}{k} \ 0.25^k \ 0.75^{16-k}=0.40498711 \end{aligned}

\displaystyle \begin{aligned} P(Y_4 \le 10)&=1-P(Y_4>10) \\&=1-\sum \limits_{j=0}^{3} \ \binom{14}{j} \ 0.25^j \ 0.75^{14-j} \\&=1-0.52133996 \\&=0.47866004 \end{aligned}

Example 5 (the above examples continued)
What is the median number of boxes of cereal purchased by Family A in order to obtain 4 boxes with the prize of free ticket to the zoo? What is the median number of boxes of cereal with undesirable prizes that are purchased by Family A?

We have the following probabilities.

$\displaystyle P(X_4>14)=\sum \limits_{k=0}^{3} \ \binom{14}{k} \ 0.25^k \ 0.75^{16-k}=0.52133996$

$\displaystyle P(X_4>15)=\sum \limits_{k=0}^{3} \ \binom{15}{k} \ 0.25^k \ 0.75^{16-k}=0.461286876$

$\displaystyle P(X_4 \le 14)=1-0.52133996=0.47866004$

$\displaystyle P(X_4 \le 15)=1-0.461286876=0.538713124$

This means that the median number of boxes to be purchased is 15. One way to look at it is that $x=$ 15 is the first number such that $P(X_4 \le x)$ is greater than 0.5. Then the median number of boxes with undesirable prizes is 11 (15 less 4).

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Comparing with the Gamma distribution

The thought process discussed in (7) and (8) certainly gives a more efficient way to calculate the cumulative distribution function and the survival function of the negative binomial distribution. Even though the negative binomial cdf can be calculated easily by software, the ideas in (7) and (8) provides a formulation that gives more insight on the negative binomial distribution.

The though process in (7) and (8) is analogous to the relationship between the Gamma distribution and the Poisson distribution. Consider the Gamma distribution where the shape parameter $n$ is a positive integer and the rate parameter $\beta$ can be any positive real number. Then the following is the density function of the Gamma distribution under consideration:

$\displaystyle f(x)=\frac{\beta^n}{(n-1)!} \ x^{n-1} \ e^{-\beta \ x} \ \ \ \ \ x>0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (9)$

The Gamma distribution described by the density function in (9) can be interpreted as a waiting time – the waiting time until the $n$th change in a Poisson process. Thus, if the $n$th change takes place after time $t$, there can be at most $n-1$ arrivals in the time interval $[0,t]$. Thus the survival function of this Gamma distribution is the same as the cdf of a Poisson distribution. The survival function in (7) is analogous to the following relation.

$\displaystyle P(X>t)=\int_t^\infty \frac{\beta^n}{(n-1)!} \ x^{n-1} \ e^{-\beta x} \ dx=\sum \limits_{j=0}^{n-1} \frac{e^{-\beta t} \ (\beta t)^j}{j!} \ \ \ \ \ \ \ \ \ \ \ (10)$

The idea for (7) and (8) is the waiting time until the $r$th success where each success or failure is based on a Bernoulli process. The resulting probability distribution is a discrete waiting time process. The idea for (10) is the waiting time until the $n$th change where each change is based on a Poisson counting process. The resulting probability distribution is a continuous waiting time process. It is not necessary to memorize these formulas. It is easy to reproduce (7), (8) and (10) from the underlying thought processes.

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$\copyright \ \text{2015 by Dan Ma}$

# Deriving some facts of the negative binomial distribution

The previous post called The Negative Binomial Distribution gives a fairly comprehensive discussion of the negative binomial distribution. In this post, we fill in some of the details that are glossed over in that previous post. We derive the following points:

• Discuss the several versions of the negative binomial distribution.
• The negative binomial probabilities sum to one, i.e., the negative binomial probability function is a valid one.
• Derive the moment generating function of the negative binomial distribution.
• Derive the first and second moments and the variance of the negative binomial distribution.
• An observation about independent sum of negative binomial distributions.

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Three versions

The negative binomial distribution has two parameters $r$ and $p$, where $r$ is a positive real number and $0. The first two versions arise from the case that $r$ is a positive integer, which can be interpreted as the random experiment of a sequence of independent Bernoulli trials until the $r$th success (the trials have the same probability of success $p$). In this interpretation, there are two ways of recording the random experiment:

$X =$ the number of Bernoulli trials required to get the $r$th success.
$Y =$ the number of Bernoulli trials that end in failure before getting the $r$th success.

The other parameter $p$ is the probability of success in each Bernoulli trial. The notation $\binom{m}{n}$ is the binomial coefficient where $m$ and $n$ are non-negative integers and $m \ge n$ is defined as:

$\displaystyle \binom{m}{n}=\frac{m!}{n! \ (m-n)!}=\frac{m(m-1) \cdots (m-(n-1))}{n!} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)$

With this in mind, the following are the probability functions of the random variables $X$ and $Y$.

$\displaystyle P(X=x)= \binom{x-1}{r-1} p^r (1-p)^{x-r} \ \ \ \ \ \ \ x=r,r+1,r+2,\cdots \ \ \ \ \ \ \ (1)$

$\displaystyle P(Y=y)=\binom{y+r-1}{y} p^r (1-p)^y \ \ \ \ \ \ \ y=0,1,2,\cdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$

The thought process for (1) is that for the event $X=x$ to happen, there can only be $r-1$ successes in the first $x-1$ trials and one additional success occurring in the last trial (the $x$th trial). The thought process for (2) is that for the event $Y=y$ to happen, there are $y+r$ trials ($y$ failures and $r$ successes). In the first $y+r-1$ trials, there can be only $y$ failures (or equivalently $r-1$ successes). Note that $X=Y+r$. Thus knowing the mean of $Y$ will derive the mean of $X$, a fact we will use below.

Instead of memorizing the probability functions (1) and (2), it is better to understand and remember the thought processes involved. Because of the natural interpretation of performing Bernoulli trials until the $r$th success, it is a good idea to introduce the negative binomial distribution via the distributions described by (1) and (2), i.e., the case where the parameter $r$ is a positive integer. When $r=1$, the random experiment is a sequence of independent Bernoulli trials until the first success (this is called the geometric distribution).

Of course, (1) and (2) can also simply be used as counting distributions without any connection with a series of Bernoulli trials (e.g. used in an insurance context as the number of losses or claims arising from a group of insurance policies).

The binomial coefficient in (0) is defined when both numbers are non-negative integers and that the top one is greater than or equal to the bottom one. However, the rightmost term in (0) can be calculated even when the top number $m$ is not a non-negative integer. Thus when $m$ is any real number, the rightmost term (0) can be calculated provided that the bottom number $n$ is a positive integer. For convenience we define $\binom{m}{0}=1$. With this in mind, the binomial coefficient $\binom{m}{n}$ is defined for any real number $m$ and any non-negative integer $n$.

The third version of the negative binomial distribution arises from the relaxation of the binomial coefficient $\binom{m}{n}$ just discussed. With this in mind, the probability function in (2) can be defined for any positive real number $r$:

$\displaystyle P(Y=y)=\binom{y+r-1}{y} p^r (1-p)^y \ \ \ \ \ \ \ y=0,1,2,\cdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$

where $\displaystyle \binom{y+r-1}{y}=\frac{(y+r-1)(y+r-2) \cdots (r+1)r}{y!}$.

Of course when $r$ is a positive integer, versions (2) and (3) are identical. When $r$ is a positive real number but is not an integer, the distribution cannot be interpreted as the number of failures until the occurrence of $r$th success. Instead, it is used as a counting distribution.

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The probabilities sum to one

Do the probabilities in (1), (2) or (3) sum to one? For the interpretations of (1) and (2), is it possible to repeatedly perform Bernoulli trials and never get the $r$th success? For $r=1$, is it possible to never even get a success? In tossing a fair coin repeatedly, soon enough you will get a head and even if $r$ is a large number, you will eventually get $r$ number of heads. Here we wish to prove this fact mathematically.

To show that (1), (2) and (3) are indeed probability functions, we use a fact concerning Maclaurin’s series expansion of the function $(1-x)^{-r}$, a fact that is covered in a calculus course. In the following two results, $r$ is a fixed positive real number and $y$ is any non-negative integer:

$\displaystyle \binom{y+r-1}{y}=(-1)^y \ \binom{-r}{y} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)$

$\displaystyle \sum \limits_{y=0}^\infty (-1)^y \ \binom{-r}{y} \ x^y=(1-x)^{-r} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)$

The result (4) is to rearrange the binomial coefficient in probability function (3) to another binomial coefficient with a negative number. This is why there is the word “negative” in negative binomial distribution. The result (5) is the Maclaurin’s series expansion for the function $(1-x)^{-r}$. We first derive these two facts and then use them to show that the negative binomial probabilities in (3) sum to one. The following derives (4).

\displaystyle \begin{aligned} \binom{y+r-1}{y}&=\frac{(y+r-1)(y+r-2) \cdots (r+1)r}{y!} \\&=(-1)^y \ \frac{(-r)(-r-1) \cdots (-r-(y-1))}{y!} \\&=(-1)^y \ \binom{-r}{y} \end{aligned}

To derive (5), let $f(x)=(1-x)^{-r}$. Based on a theorem that can be found in most calculus text, the function $f(x)$ has the following Maclaurin’s series expansion (Maclaurin’s series is simply Taylor’s series with center = 0).

$\displaystyle (1-x)^{-r}=f(0)+f^{'}(0)x+\frac{f^{(2)}(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+\cdots + \frac{f^{(n)}(0)}{n!}x^n+\cdots$

where $-1. Now, filling in the derivatives $f^{(n)}(0)$, we have the following derivation.

\displaystyle \begin{aligned} (1-x)^{-r}&=1+rx+\frac{(r+1)r}{2!}x^2+\frac{(r+2)(r+1)r}{3!}x^3 \\& \ \ \ \ \ \ \ \ +\cdots+\frac{(r+y-1)(r+y-2) \cdots (r+1)r}{y!}x^y +\cdots \\&=1+(-1)^1 (-r)x+(-1)^2\frac{(-r)(-r-1)}{2!}x^2 \\& \ \ \ \ \ \ +(-1)^3 \frac{(-r)(-r-1)(-r-2)}{3!}x^3 +\cdots \\& \ \ \ \ \ \ +(-1)^y \frac{(-r)(-r-1) \cdots (-r-y+2)(-r-y+1)}{y!}x^y +\cdots \\&=(-1)^0 \binom{-r}{0}x^0 +(-1)^1 \binom{-r}{1}x^1+(-1)^2 \binom{-r}{2}x^2 \\& \ \ \ \ \ \ +(-1)^3 \binom{-r}{3}x^3+\cdots +(-1)^y \binom{-r}{y}x^y+\cdots \\&=\sum \limits_{y=0}^\infty (-1)^y \ \binom{-r}{y} \ x^y \end{aligned}

We can now show that the negative binomial probabilities in (3) sum to one. Let $q=1-p$.

\displaystyle \begin{aligned} \sum \limits_{y=0}^\infty \binom{y+r-1}{y} \ p^r \ q^y &=p^r \ \sum \limits_{y=0}^\infty (-1)^y \ \binom{-r}{y} \ q^y \ \ \ \ \ \ \ \ \ \ \ \text{using } (4) \\&=p^r \ (1-q)^{-r} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{using } (5)\\&=p^r p^{-r} \\&=1 \end{aligned}

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The moment generating function

We now derive the moment generating function of the negative binomial distribution according to (3). The moment generation function is $M(t)=E(e^{tY})$ over all real numbers $t$ for which $M(t)$ is defined. The following derivation does the job.

\displaystyle \begin{aligned} M(t)&=E(e^{tY}) \\&=\sum \limits_{y=0}^\infty \ e^{t y} \ \binom{y+r-1}{y} \ p^r \ (1-p)^y \\&=p^r \ \sum \limits_{y=0}^\infty \ \binom{y+r-1}{y} \ [(1-p) e^t]^y \\&=p^r \ \sum \limits_{y=0}^\infty \ (-1)^y \binom{-r}{y} \ [(1-p) e^t]^y \ \ \ \ \ \ \ \ \ \ \ \text{using } (4) \\&=p^r \ [1-(1-p) \ e^t]^{-r} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{using } (5) \\&=\frac{p^r}{[1-(1-p) \ e^t]^{r}}\end{aligned}

The above moment generating function works for the negative binomial distribution with respect to (3) and thus to (2). For the distribution in (1), note that $X=Y+r$. Thus $E(e^{tX})=E(e^{t(Y+r)})=e^{tr} \ E(e^{tY})$. The moment generating function of (1) is simply the above moment generating function multiplied by the factor $e^{tr}$. To summarize, the moment generating functions for the three versions are:

$\displaystyle M_X(t)=E[e^{tX}]=\frac{p^r \ e^{tr}}{[1-(1-p) \ e^t]^{r}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{for } (1)$

$\displaystyle M_Y(t)=E[e^{tY}]=\frac{p^r}{[1-(1-p) \ e^t]^{r}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{for } (2) \text{ and } (3)$

The domain of the moment generating function is the set of all $t$ that for which $M_X(t)$ or $M_Y(t)$ is defined and is positive. Based on the form that it takes, we focus on making sure that $1-(1-p) \ e^t>0$. This leads to the domain $t<-\text{ln}(1-p)$.

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The mean and the variance

With the moment generating function derived in the above section, we can now focus on finding the moments of the negative binomial distribution. To find the moments, simply take the derivatives of the moment generating function and evaluate at $t=0$. For the distribution represented by the probability function in (3), we calculate the following:

$E(Y)=M_Y^{'}(0)$

$E(Y^2)=M_Y^{(2)}(0)$

$Var(Y)=E(Y^2)-E(Y)^2$

After taking the first and second derivatives and evaluate at $t=0$, the first and the second moments are:

$\displaystyle E(Y)=r \ \frac{1-p}{p}$

$\displaystyle E(Y^2)=\frac{r(1-p)[1+(1-p)]}{p^2}$

The following derives the variance.

\displaystyle \begin{aligned} Var(Y)&=E(Y^2)-E(Y)^2 \\&=\frac{r(1-p)[1+(1-p)]}{p^2}-\frac{(1-p)^2}{p^2} \\&=\frac{r(1-p)[1+r(1-p)-r(1-p)]}{p^2} \\&=\frac{r(1-p)}{p^2} \end{aligned}

The above formula is the variance for the three versions (1), (2) and (3). Note that $Var(Y)>E(Y)$. In contrast, the variance of the Poisson distribution is identical to its mean. Thus in the situation where the variance of observed data is greater than the sample mean, the negative binomial distribution should be a better fit than the Poisson distribution.

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The independent sum

There is an easy consequence that follows from the moment generating function derived above. The sum of several independent negative binomial distributions is also a negative binomial distribution. For example, suppose $T_1,T_2, \cdots,T_n$ are independent negative binomial random variables (version (3)). Suppose each $T_j$ has parameters $r_j$ and $p$ (the second parameter is identical). The moment generating function of the independent sum is the product of the individual moment generating functions. Thus the following is the moment generating function of $T=T_1+\cdots+T_n$.

$\displaystyle M_T(t)=E[e^{tT}]=\frac{p^g}{[1-(1-p) \ e^t]^{g}}$

where $g=r_1+\cdots+r_n$. The moment generating function uniquely identifies the distribution. The above $M_T(t)$ is that of a negative binomial distribution with parameters $g$ and $p$ according to (3).

A special case is that the sum of $n$ independent geometric distributions is a negative binomial distribution with the $r$ parameter being $r=n$. The following is the moment generating function of the sum $W$ of $n$ independent geometric distributions.

$\displaystyle M_W(t)=E[e^{tW}]=\frac{p^n}{[1-(1-p) \ e^t]^{n}}$

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$\copyright \ \text{2015 by Dan Ma}$

# Conditional Distributions, Part 2

We present more examples to further illustrate the thought process of conditional distributions. A conditional distribution is a probability distribution derived from a given probability distribution by focusing on a subset of the original sample space (we assume that the probability distribution being discussed is a model for some random experiment). The new sample space (the subset of the original one) may be some outcomes that are of interest to an experimenter in a random experiment or may reflect some new information we know about the random experiment in question. We illustrate this thought process in the previous post Conditional Distributions, Part 1 using discrete distributions. In this post, we present some continuous examples for conditional distributions. One concept illustrated by the examples in this post is the notion of mean residual life, which has an insurance interpretation (e.g. the average remaining time until death given that the life in question is alive at a certain age).

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The Setting

The thought process of conditional distributions is discussed in the previous post Conditional Distributions, Part 1. We repeat the same discussion using continuous distributions.

Let $X$ be a continuous random variable that describes a certain random experiment. Let $S$ be the sample space of this random experiment. Let $f(x)$ be its probability density function.

We assume that $X$ is a univariate random variable, meaning that the sample space $S$ is the real line $\mathbb{R}$ or a subset of $\mathbb{R}$. Since $X$ is a continuous random variable, we know that $S$ would contain an interval, say, $(a,b)$.

Suppose that in the random experiment in question, certain event $A$ has occurred. The probability of the event $A$ is obtained by integrating the density function over the set $A$.

$\displaystyle P(A)=\int_{x \in A} f(x) \ dx$

Since the event $A$ has occurred, $P(A)>0$. Since we are dealing with a continuous distribution, the set $A$ would contain an interval, say $(c,d)$ (otherwise $P(A)=0$). So the new probability distribution we define is also a continuous distribution. The following is the density function defined on the new sample space $A$.

$\displaystyle f(x \lvert A)=\frac{f(x)}{P(A)}, \ \ \ \ \ \ \ \ \ x \in A$

The above probability distribution is called the conditional distribution of $X$ given the event $A$, denoted by $X \lvert A$. This new probability distribution incorporates new information about the results of a random experiment.

Once this new probability distribution is established, we can compute various distributional quantities (e.g. cumulative distribution function, mean, variance and other higher moments).

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Examples

Example 1

Let $X$ be the lifetime (in years) of a brand new computer purchased from a certain manufacturer. Suppose that the following is the density function of the random variable $X$.

$\displaystyle f(x)=\frac{3}{2500} \ (100x-20x^2 + x^3), \ \ \ \ \ \ \ \ 0

Suppose that you have just purchased a one such computer that is 2-year old and in good working condition. We have the following questions.

• What is the expected lifetime of this 2-year old computer?
• What is the expected number of years of service that will be provided by this 2-year old computer?

Both calculations are conditional means since the computer in question already survived to age 2. However, there is a slight difference between the two calculations. The first one is the expected age of the 2-year old computer, i.e., the conditional mean $E(X \lvert X>2)$. The second one is the expected remaining lifetime of the 2-year old computer, i.e., $E(X-2 \lvert X>2)$.

For a brand new computer, the sample space is the interval $S=0. Knowing that the computer is already 2-year old, the new sample space is $A=2. The total probability of the new sample space is:

$\displaystyle P(A)=P(X>2)=\int_{2}^{10} \frac{3}{2500} \ (100x-20x^2 + x^3) \ dx=\frac{2048}{2500}=0.8192$

The conditional density function of $X$ given $X>2$ is:

\displaystyle \begin{aligned} f(x \lvert X>2)&=\frac{\frac{3}{2500} \ (100x-20x^2 + x^3)} {\frac{2048}{2500}} \\&=\frac{3}{2048} \ (100x-20x^2 + x^3), \ \ \ \ \ \ \ \ \ 2

The first conditional mean is:

\displaystyle \begin{aligned} E(X \lvert X>2)&=\int_2^{10} x \ f(x \lvert X>2) \ dx \\&=\int_2^{10} \frac{3}{2048} \ x(100x-20x^2 + x^3) \ dx \\&=\int_2^{10} \frac{3}{2048} \ (100x^2-20x^3 + x^4) \ dx \\&=\frac{47104}{10240}=4.6 \end{aligned}

The second conditional mean is:

$\displaystyle E(X-2 \lvert X>2)=E(X \lvert X>2)-2=2.6$

In contrast, the unconditional mean is:

$\displaystyle E(X)=\int_0^{10} \frac{3}{2500} \ (100x^2-20x^3 + x^4) \ dx=4$

So if the lifetime of a computer is modeled by the density function $f(x)$ given here, the expected lifetime of a brand new computer is 4 years. If you know that a computer has already been in use for 2 years and is in good condition, the expected lifetime is 4.6 years, where 2 years of which have already passed, showing us that the remaining lifetime is 2.6 years.

Note that the following calculation is not $E(X \lvert X>2)$, though is something that some students may attempt to do.

$\displaystyle \int_2^{10} x \ f(x) \ dx =\int_2^{10} \frac{3}{2500} \ x(100x-20x^2 + x^3) \ dx=\frac{47104}{12500}=3.76832$

The above calculation does not use the conditional distribution that $X>2$. Also note that the answer is less than the unconditional mean $E(X)$.

Example 2 – Exponential Distribution

Work Example 1 again by assuming that the lifetime of the type of computers in questions follows the exponential distribution with mean 4 years.

The following is the density function of the lifetime $X$.

$\displaystyle f(x)=0.25 \ e^{-0.25 x}, \ \ \ \ \ \ 0

The probability that the computer has survived to age 2 is:

$\displaystyle P(X>2)=\int_2^\infty 0.25 \ e^{-0.25 x} \ dx=e^{-0.25 (2)}=e^{-0.5}$

The conditional density function given that $X>2$ is:

$\displaystyle f(x \lvert X>2)= \frac{0.25 \ e^{-0.25 x}}{e^{-0.25 (2)}}=0.25 \ e^{-0.25 (x-2)}, \ \ \ \ \ \ \ 2

To compute the conditional mean $E(X \lvert X>2)$, we have

\displaystyle \begin{aligned} E(X \lvert X>2)&=\int_2^\infty x \ f(x \lvert X>2) \ dx \\&=\int_2^\infty 0.25 \ x \ e^{-0.25 (x-2)} \ dx \\&=\int_0^\infty 0.25 \ (u+2) \ e^{-0.25 u} \ du \ \ \ (\text{change of variable}) \\&=\int_0^\infty 0.25 \ u \ e^{-0.25 u} \ du+2\int_0^\infty 0.25 \ e^{-0.25 u} \ du \\&=\frac{1}{0.25}+2=4+2=6\end{aligned}

Then $\displaystyle E(X-2 \lvert X>2)=E(X \lvert X>2)-2=6-2=4$.

We have an interesting result here. The expected lifetime of a brand new computer is 4 years. Yet the remaining lifetime for a 2-year old computer is still 4 years! This is the no-memory property of the exponential distribution – if the lifetime of a type of machines is distributed according to an exponential distribution, it does not matter how old the machine is, the remaining lifetime is always the same as the unconditional mean! This point indicates that the exponential distribution is not an appropriate for modeling the lifetime of machines or biological lives that wear out over time.

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Mean Residual Life

If a 40-year old man who is a non-smoker wants to purchase a life insurance policy, the insurance company is interested in knowing the expected remaining lifetime of the prospective policyholder. This information will help determine the pricing of the life insurance policy. The expected remaining lifetime of the prospective policyholder is called is called the mean residual life and is the conditional mean $E(X-t \lvert X>t)$ where $X$ is a model for the lifetime of some life.

In engineering and manufacturing applications, probability modeling of lifetimes of objects (e.g. devices, systems or machines) is known as reliability theory. The mean residual life also plays an important role in such applications.

Thus if the random variable $X$ is a lifetime model (lifetime of a life, system or device), then the conditional mean $E(X-t \lvert X>t)$ is called the mean residual life and is the expected remaining lifetime of the life or system in question given that the life has survived to age $t$.

On the other hand, if the random variable $X$ is a model of insurance losses, then the conditional mean $E(X-t \lvert X>t)$ is the expected claim payment per loss given that the loss has exceeded the deductible of $t$. In this interpretation, the conditional mean $E(X-t \lvert X>t)$ is called the mean excess loss function.

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Summary

In conclusion, we summarize the approach for calculating the two conditional means demonstrated in the above examples.

Suppose $X$ is a continuous random variable with the support being $(0,\infty)$ (the positive real numbers), with $f(x)$ being the density function. The following is the density function of the conditional probability distribution given that $X>t$.

$\displaystyle f(x \lvert X>t)=\frac{f(x)}{P(X>t)}, \ \ \ \ \ \ \ \ \ x>t$

Then we have the two conditional means:

$\displaystyle E(X \lvert X>t)=\int_t^\infty x \ f(x \lvert X>t) \ dx=\int_t^\infty x \ \frac{f(x)}{P(X>t)} \ dx$

$\displaystyle E(X-t \lvert X>t)=\int_t^\infty (x-t) \ f(x \lvert X>t) \ dx=\int_t^\infty (x-t) \ \frac{f(x)}{P(X>t)} \ dx$

If $E(X \lvert X>t)$ is calculated first (or is easier to calculate), then $E(X-t \lvert X>t)=E(X \lvert X>t)-t$, as shown in the above examples.

If $X$ is a discrete random variable, then the integrals are replaced by summation symbols. As indicated above, the conditional mean $E(X-t \lvert X>t)$ is called the mean residual life when $X$ is a probability model of the lifetime of some system or life.

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Practice Problems

Practice problems are found in the companion blog.

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$\copyright \ \text{2013 by Dan Ma}$

# Conditional Distributions, Part 1

We illustrate the thought process of conditional distributions with a series of examples. These examples are presented in a series of blog posts. In this post, we look at some conditional distributions derived from discrete probability distributions.

Practice problems are found in the companion blog.

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The Setting

Suppose we have a discrete random variable $X$ with $f(x)=P(X=x)$ as the probability mass function. Suppose some random experiment can be modeled by the discrete random variable $X$. The sample space $S$ for this probability experiment is the set of sample points with positive probability masses, i.e. $S$ is the set of all $x$ for which $f(x)=P(X=x)>0$. In the examples below, $S$ is either a subset of the real line $\mathbb{R}$ or a subset of the plane $\mathbb{R} \times \mathbb{R}$. Conceivably the sample space could be subset of any Euclidean space $\mathbb{R}^n$ in higher dimension.

Suppose that we are informed that some event $A$ in the random experiment has occurred ($A$ is a subset of the sample space $S$). Given this new information, all the sample points outside of the event $A$ are irrelevant. Or perhaps, in this random experiment, we are only interested in those outcomes that are elements of some subset $A$ of the sample space $S$. In either of these scenarios, we wish to make the event $A$ as a new sample space.

The probability of the event $A$, denoted by $P(A)$, is derived by summing the probabilities $f(x)=P(X=x)$ over all the sample points $x \in A$. We have:

$\displaystyle P(A)=\sum_{x \in A} P(X=x)$

The probability $P(A)$ may not be 1.0. So the probability masses $f(x)=P(X=x)$ for the sample points $x \in A$, if they are unadjusted, may not form a probability distribution. However, if we consider each such probability mass $f(x)=P(X=x)$ as a proportion of the probability $P(A)$, then the probability masses of the event $A$ will form a probability distribution. For example, say the event $A$ consists of two probability masses 0.2 and 0.3, which sum to 0.5. Then in the new sample space, the first probability mass is 0.4 (0.2 multiplied by $\displaystyle \frac{1}{0.5}$ or divided by 0.5) and the second probability mass is 0.6.

We now summarize the above paragraph. Using the event $A$ as a new sample space, the probability mass function is:

$\displaystyle f(x \lvert A)=\frac{f(x)}{P(A)}=\frac{P(X=x)}{P(A)}, \ \ \ \ \ \ \ \ \ x \in A$

The above probability distribution is called the conditional distribution of $X$ given the event $A$, denoted by $X \lvert A$. This new probability distribution incorporates new information about the results of a random experiment.

Once this new probability distribution is established, we can compute various distributional quantities (e.g. cumulative distribution function, mean, variance and other higher moments).

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Examples

Suppose that two students take a multiple choice test that has 5 questions. Let $X$ be the number of correct answers of one student and $Y$ be the number of correct answers of the other student (these can be considered as test scores for the purpose of the examples here). Assume that $X$ and $Y$ are independent. The following shows the probability functions.

$\displaystyle \begin{bmatrix} \text{Count of}&\text{ }&\text{ }&P(X=x) &\text{ }&\text{ }&P(Y=y) \\\text{Correct Answers}&\text{ }&\text{ }&\text{ } &\text{ }&\text{ }&\text{ } \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 0&\text{ }&\text{ }&0.4&\text{ }&\text{ }&0.1 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 1&\text{ }&\text{ }&0.2&\text{ }&\text{ }&0.1 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 2&\text{ }&\text{ }&0.1&\text{ }&\text{ }&0.2 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 3&\text{ }&\text{ }&0.1&\text{ }&\text{ }&0.2 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 4&\text{ }&\text{ }&0.1 &\text{ }&\text{ }&0.2 \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 5&\text{ }&\text{ }&0.1 &\text{ }&\text{ }&0.2 \end{bmatrix}$

Note that $E(X)=1.6$ and $E(Y)=2.9$. Without knowing any additional information, we can expect on average one student gets 1.6 correct answers and one student gets 2.9 correct answers. If having 3 or more correct answers is considered passing, then the student represented by $X$ has a 30% chance of passing while the student represented by $Y$ has a 60% chance of passing. The following examples show how the expectation can change as soon as new information is known.

The following examples are based on these two test scores $X$ and $Y$.

Example 1

In this example, we only consider the student whose correct answers are modeled by the random variable $X$. In addition to knowing the probability function $P(X=x)$, we also know that this student has at least one correct answer (i.e. the new information is $X>0$).

In light of the new information, the new sample space is $A=\left\{1,2,3,4,5 \right\}$. Note that $P(A)=0.6$. In this new sample space, each probability mass is the original one divided by 0.6. For example, for the sample point $X=1$, we have $\displaystyle P(X=1 \lvert X>0)=\frac{0.2}{0.6}=\frac{2}{6}$. The following is the conditional probability distribution of $X$ given $X>0$.

$\displaystyle P(X=1 \lvert X>0)=\frac{2}{6}$

$\displaystyle P(X=2 \lvert X>0)=\frac{1}{6}$

$\displaystyle P(X=3 \lvert X>0)=\frac{1}{6}$

$\displaystyle P(X=4 \lvert X>0)=\frac{1}{6}$

$\displaystyle P(X=5 \lvert X>0)=\frac{1}{6}$

The conditional mean is the mean of the conditional distribution. We have $\displaystyle E(X \lvert X>0)=\frac{16}{6}=2.67$. Given that this student is knowledgeable enough to answer some question correctly, the expectation is higher than before knowing the additional information. Also, given the new information, the student in question has a 50% chance of passing (vs. 30% before the new information is known).

Example 2

We now look at a joint distribution that has a 2-dimensional sample space. Consider the joint distribution of test scores $X$ and $Y$. If the new information is that the total number of correct answers among them is 4, how would this change our expectation of their performance?

Since $X$ and $Y$ are independent, the sample space is a square as indicated the figure below.

$\text{ }$

Figure 1 – Sample Space of Test Scores

Because the two scores are independent, the joint probability at each of these 36 sample points is the product of the individual probabilities. We have $P(X=x,Y=y)=P(X=x) \times P(Y=y)$. The following figure shows one such joint probability.

Figure 2 – Joint Probability Function

After taking the test, suppose that we have the additional information that the two students have a total of 4 correct answers. With this new information, we can focus our attention on the new sample space that is indicated in the following figure.

Figure 3 – New Sample Space

Now we wish to discuss the conditional probability distribution of $X \lvert X+Y=4$ and the conditional probability distribution of $Y \lvert X+Y=4$. In particular, given that there are 4 correct answers between the two students, what would be their expected numbers of correct answers and what would be their chances of passing?

There are 5 sample points in the new sample space (the 5 points circled above). The conditional probability distribution is obtained by making each probability mass as a fraction of the sum of the 5 probability masses. First we calculate the 5 joint probabilities.

$\displaystyle P(X=0,Y=4)=P(X=0) \times P(Y=4) =0.4 \times 0.2=0.08$

$\displaystyle P(X=1,Y=3)=P(X=1) \times P(Y=3) =0.2 \times 0.2=0.04$

$\displaystyle P(X=2,Y=2)=P(X=2) \times P(Y=2) =0.1 \times 0.2=0.02$

$\displaystyle P(X=3,Y=1)=P(X=3) \times P(Y=1) =0.1 \times 0.1=0.01$

$\displaystyle P(X=4,Y=0)=P(X=4) \times P(Y=0) =0.1 \times 0.1=0.01$

The sum of these 5 joint probabilities is $P(X+Y=4)=0.16$. Making each of these joint probabilities as a fraction of 0.16, we have the following two conditional probability distributions.

$\displaystyle P(X=0 \lvert X+Y=4)=\frac{8}{16} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P(Y=0 \lvert X+Y=4)=\frac{1}{16}$

$\displaystyle P(X=1 \lvert X+Y=4)=\frac{4}{16} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P(Y=1 \lvert X+Y=4)=\frac{1}{16}$

$\displaystyle P(X=2 \lvert X+Y=4)=\frac{2}{16} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P(Y=2 \lvert X+Y=4)=\frac{2}{16}$

$\displaystyle P(X=3 \lvert X+Y=4)=\frac{1}{16} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P(Y=3 \lvert X+Y=4)=\frac{4}{16}$

$\displaystyle P(X=4 \lvert X+Y=4)=\frac{1}{16} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P(Y=4 \lvert X+Y=4)=\frac{8}{16}$

Now the conditional means given that $X+Y=4$, comparing against the unconditional means.

$\displaystyle E(X \lvert X+Y=4)=\frac{0+4+4+3+4}{16}=\frac{15}{16}=0.9375 \ \ \ \ \ \ \ \ \text{vs} \ \ E(X)=1.6$

$\displaystyle E(Y \lvert X+Y=4)=\frac{0+1+4+12+32}{16}=\frac{49}{16}=3.0625 \ \ \ \ \ \text{vs} \ \ E(Y)=2.9$

Now compare the chances of passing.

$\displaystyle P(X \ge 3 \lvert X+Y=4)=\frac{4}{16}=0.25 \ \ \ \ \ \ \ \ \ \ \text{vs} \ \ P(X \ge 3)=0.3$

$\displaystyle P(Y \ge 3 \lvert X+Y=4)=\frac{14}{16}=0.875 \ \ \ \ \ \ \ \ \text{vs} \ \ P(Y \ge 3)=0.6$

Based on the new information of $X+Y=4$, we have a lower expectation for the student represented by $X$ and a higher expectation for the student represented by $Y$. Observe that the conditional probability at $X=0$ increases to 0.5 from 0.4, while the conditional probability at $X=4$ increases to 0.5 from 0.2.

Example 3

Now suppose the new information is that the two students do well on the test. Particularly, their combined number of correct answers is greater than or equal to 5, i.e., $X+Y \ge 5$. How would this impact the conditional distributions?

First we discuss the conditional distributions for $X \lvert X+Y \ge 5$ and $Y \lvert X+Y \ge 5$. By considering the new information, the following is the new sample space.

Figure 4 – New Sample Space

To derive the conditional distribution of $X \lvert X+Y \ge 5$, sum the joint probabilities within the new sample space for each $X=x$. The calculation is shown below.

$\displaystyle P(X=0 \cap X+Y \ge 5)=0.4 \times 0.2=0.08$

$\displaystyle P(X=1 \cap X+Y \ge 5)=0.2 \times (0.2+0.2)=0.08$

$\displaystyle P(X=2 \cap X+Y \ge 5)=0.1 \times (0.2+0.2+0.2)=0.06$

$\displaystyle P(X=3 \cap X+Y \ge 5)=0.1 \times (0.2+0.2+0.2+0.2)=0.08$

$\displaystyle P(X=4 \cap X+Y \ge 5)=0.1 \times (1-0.1)=0.09$

$\displaystyle P(X=5 \cap X+Y \ge 5)=0.1 \times (1)=0.10$

The sum of these probabilities is 0.49, which is $P(X+Y \ge 5)$. The conditional distribution of $X \lvert X+Y \ge 5$ is obtained by taking each of the above probabilities as a fraction of 0.49. We have:

$\displaystyle P(X=0 \lvert X+Y \ge 5)=\frac{8}{49}=0.163$

$\displaystyle P(X=1 \lvert X+Y \ge 5)=\frac{8}{49}=0.163$

$\displaystyle P(X=2 \lvert X+Y \ge 5)=\frac{6}{49}=0.122$

$\displaystyle P(X=3 \lvert X+Y \ge 5)=\frac{8}{49}=0.163$

$\displaystyle P(X=4 \lvert X+Y \ge 5)=\frac{9}{49}=0.184$

$\displaystyle P(X=5 \lvert X+Y \ge 5)=\frac{10}{49}=0.204$

We have the conditional mean $\displaystyle E(X \lvert X+Y \ge 5)=\frac{0+8+12+24+36+50}{49}=\frac{130}{49}=2.653$ (vs. $E(X)=1.6$). The conditional probability of passing is $\displaystyle P(X \ge 3 \lvert X+Y \ge 5)=\frac{27}{49}=0.55$ (vs. $P(X \ge 3)=0.3$).

Note that the above conditional distribution for $X \lvert X+Y \ge 5$ is not as skewed as the original one for $X$. With the information that both test takers do well, the expected score for the student represented by $X$ is much higher.

With similar calculation we have the following results for the conditional distribution of $Y \lvert X+Y \ge 5$.

$\displaystyle P(Y=0 \lvert X+Y \ge 5)=\frac{1}{49}=0.02$

$\displaystyle P(Y=1 \lvert X+Y \ge 5)=\frac{2}{49}=0.04$

$\displaystyle P(Y=2 \lvert X+Y \ge 5)=\frac{6}{49}=0.122$

$\displaystyle P(Y=3 \lvert X+Y \ge 5)=\frac{8}{49}=0.163$

$\displaystyle P(Y=4 \lvert X+Y \ge 5)=\frac{12}{49}=0.245$

$\displaystyle P(Y=5 \lvert X+Y \ge 5)=\frac{20}{49}=0.408$

We have the conditional mean $\displaystyle E(Y \lvert X+Y \ge 5)=\frac{0+2+12+24+48+100}{49}=\frac{186}{49}=3.8$ (vs. $E(Y)=2.9$). The conditional probability of passing is $\displaystyle P(Y \ge 3 \lvert X+Y \ge 5)=\frac{40}{49}=0.82$ (vs. $P(Y \ge 3)=0.6$). Indeed, with the information that both test takers do well, we can expect much higher results from each individual test taker.

Example 4

In Examples 2 and 3, the new information involve both test takers (both random variables). If the new information involves just one test taker, it may be immaterial on the exam score of the other student. For example, suppose that $Y \ge 4$. Then what is the conditional distribution for $X \lvert Y \ge 4$? Since $X$ and $Y$ are independent, the high score $Y \ge 4$ has no impact on the score $X$. However, the high joint score $X+Y \ge 5$ does have an impact on each of the individual scores (Example 3).

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Summary

We conclude with a summary of the thought process of conditional distributions.

Suppose $X$ is a discrete random variable and $f(x)=P(X=x)$ is its probability function. Further suppose that $X$ is the probability model of some random experiment. The sample space of this random experiment is $S$.

Suppose we have some new information that in this random experiment, some event $A$ has occurred. The event $A$ is a subset of the sample space $S$.

To incorporate this new information, the event $A$ is the new sample space. The random variable incorporated with the new information, denoted by $X \lvert A$, has a conditional probability distribution. The following is the probability function of the conditional distribution.

$\displaystyle f(x \lvert A)=\frac{f(x)}{P(A)}=\frac{P(X=x)}{P(A)}, \ \ \ \ \ \ \ \ \ x \in A$

where $P(A)$ = $\displaystyle \sum_{x \in A} P(X=x)$.

The thought process is that in the conditional distribution is derived from taking each original probability mass as a fraction of the total probability $P(A)$. The probability function derived in this manner reflects the new information that the event $A$ has occurred.

Once the conditional probability function is derived, it can be used just like any other probability function, e.g. computationally for finding various distributional quantities.

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Practice Problems

Practice problems are found in the companion blog.

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$\copyright \ \text{2013 by Dan Ma}$

# Looking for a Match

I recently gave an exam in my statistics course, which turned out to be an excellent example for the matching problem, a classic problem in probability. There were 66 students taking the test. I wrote down the names of the students in the order of turning in the exam. The following table shows the positions of the students in alphabetical order and in the order they turned in the exam.

For example, the first student in the alphabetical order was the 12th student who turned in the exam. The first student who turned in the exam was the 37th student in the alphabetical order. It turned out that there was a student who had the same position in both orders. Such a student is called a match (see the following table).

This example captures the essence of a classic problem in probability called the matching problem. There are many other colorful descriptions of the matching problem. All of them involve the pairing of two orderings on a set of objects. We have a set of objects or items that are ordered in some natural way. Then we scramble the items randomly (or according to some ordering that is unpredictable in advance). Whenever an item has the same position in both the original order and in the scrambled order, it is called a match.

In our exam example, one version of the matching problem asks: what is the probability that there is at least one student that is a match?

In fact, in matching two orderings, such as the one described here, a match happens more often than not. The probability of having at least one match is roughly 0.63. Specifically, when $n$ is the number of students taking the exam, the probability of finding at least one match approaches $1-e^{-1}=0.632121$ as $n \rightarrow \infty$. The derivation of this probability is based on the inclusion-exclusion principle and is discussed in the blog post called The Matching Problem.

Even though the probability of having at least one match is a function of $n$ (the number of items), the probability converges to $1-e^{-1}$ pretty rapidly. Thus for all practical purposes, we can say that the probability of having at least one match is 0.63 (roughly two-third), whenever the number of items involved in the random scrambling is reasonably large (as in the 66 students taking the exam).

Instead of finding the probability of having at least one match, we can also ask for the probability of having exactly $k$ matches, where $k$ is any integer from $0$ to $n$. Let $X_n$ be the number of matches when we match the “exam turning in” order with the alphabetical order for $n$ students. The probability function $P(X_n=k)$ is derived in the blog post called More About the Matching Problem.

The blog post More About the Matching Problem also points out that $P(X_n=k)$ is approximated by the Poisson distribution with parameter $\alpha=1$. Thus we have:

$\displaystyle (1) \ \ \ \ \ P(X_n=k) \approx \frac{e^{-1}}{k!} \ \ \ \ \ \ \ \ \ \ k=0,1,2,\cdots,n$

The following are the first 4 probabilities of $P(X_n=k)$.

$\displaystyle (2) \ \ \ \ \ P(X_n=0) \approx \frac{e^{-1}}{0!}=0.3679$

$\displaystyle (3) \ \ \ \ \ P(X_n=1) \approx \frac{e^{-1}}{1!}=0.3679$

$\displaystyle (4) \ \ \ \ \ P(X_n=2) \approx \frac{e^{-1}}{2!}=0.1839$

$\displaystyle (5) \ \ \ \ \ P(X_n=3) \approx \frac{e^{-1}}{3!}=0.0313$

In the random experiment of matching two orderings on the same set of objects, about 37% of the time, there is no match and about 37% of the time there is exactly one match. Having no matches and having exactly one match are the mostly scenarios (occurring about 74% of the time). Having 2 matches is possible, but only happens about 18% of the time. It is rare to have 3 ore more matches.

Another interesting observation is that if a match occurs, it is mostly likely that there is only one match (such as the example discussed here).

There are many colorful descriptions of the matching problem. The possibilities are limitless. One example is that of $n$ married couples going to a ball room dancing class. The instructor randomly assign the ladies to the gentlemen as dancing partners. A match occurs if a wife is assigned as
the dancing partner of her husband.

A previous blog post (Tis the Season for Gift Exchange) presents an example involving gift exchange. Each person attending a party brings a gift for gift exchange. The gifts are put in a pile and each person randomly selects a gift from the pile. A match occurs if a person selects his or her own gift.

In another blog post (A lazy professor who lets students do their own grading), a professor randomly returns quizzes to the students for grading. A match occurs if a students is assigned his or her own quiz.

# An Introduction to the Bayes’ Formula

We open up a discussion of the Bayes’ formula by going through a basic example. The Bayes’ formula or theorem is a method that can be used to compute “backward” conditional probabilities such as the examples described here. The formula will be stated after we examine the calculation from Example 1. The following diagram describes Example 1. Example 2 is presented at the end of the post and is left as exercise. For a basic discussion of the Bayes’ formula, see [1] and chapter 4 of [2].

Example 1

As indicated in the diagram, Box 1 has 1 red ball and three white balls and Box 2 has 2 red balls and 2 white balls. The example involves a sequence of two steps. In the first step (the green arrow in the above diagram), a box is randomly chosen from two boxes. In the second step (the blue arrow), a ball is randomly selected from the chosen box. We assume that the identity of the chosen box is unknown to the participants of this random experiment (e.g. suppose the two boxes are identical in appearance and a box is chosen by your friend and its identity is kept from you). Since a box is chosen at random, it is easy to see that $P(B_1)=P(B_2)=0.5$.

The example involves conditional probabilities. Some of the conditional probabilities are natural and are easy to see. For example, if the chosen box is Box 1, it is clear that the probability of selecting a red ball is $\displaystyle \frac{1}{4}$, i.e. $\displaystyle P(R \lvert B_1)=\frac{1}{4}$. Likewise, the conditional probability $P(R \lvert B_2)$ is $\displaystyle \frac{2}{4}$. These two conditional probabilities are “forward” conditional probabilities since the events $R \lvert B_1$ and $R \lvert B_2$ occur in a natural chronological order.

What about the reversed conditional probabilities $P(B_1 \lvert R)$ and $P(B_2 \lvert R)$? In other words, if the selected ball from the unknown box (unknown to you) is red, what is the probability that the ball is from Box 1?

The above question seems a little backward. After the box is randomly chosen, it is fixed (though the identity is unknown to you). Since it is fixed, shouldn’t the probability that the box being Box 1 is $\displaystyle \frac{1}{2}$? Since the box is already chosen, how can the identity of the box be influenced by the color of the ball selected from it? The answer is of course no.

We should not look at the chronological sequence of events. Instead, the key to understanding the example is through performing the random experiment repeatedly. Think of the experiment of choosing one box and then selecting one ball from the chosen box. Focus only on the trials that result in a red ball. For the result to be a red ball, we need to get either Box 1/ Red or Box 2/Red. Compute the probabilities of these two cases. Then add these two probabilities, we will obtain the probability that the selected ball is red. The following diagram illustrates this calculation.

Example 1 – Tree Diagram

The outcomes with red border in the above diagram are the outcomes that result in a red ball. The diagram shows that if we perform this experiment many times, about 37.5% of the trials will result in a red ball (on average 3 out of 8 trials will result in a red ball). In how many of these trials, is Box 1 the source of the red ball? In the diagram, we see that the case Box 2/Red is twice as likely as the case Box 1/Red. We conclude that the case Box 1/Red accounts for about one third of the cases when the selected ball is red. In other words, one third of the red balls come from Box 1 and two third of the red balls come from Box 2. We have:

$\displaystyle (1) \ \ \ \ \ P(B_1 \lvert R)=\frac{1}{3}$

$\displaystyle (2) \ \ \ \ \ P(B_2 \lvert R)=\frac{2}{3}$

Instead of using the tree diagram or the reasoning indicated in the paragraph after the tree diagram, we could just as easily apply the Bayes’ formula:

\displaystyle \begin{aligned}(3) \ \ \ \ \ P(B_1 \lvert R)&=\frac{P(R \lvert B_1) \times P(B_1)}{P(R)} \\&=\frac{\frac{1}{2} \times \frac{1}{4}}{\frac{3}{8}} \\&=\frac{1}{3} \end{aligned}

In the calculation in $(3)$ (as in the tree diagram), we use the law of total probability:

\displaystyle \begin{aligned}(4) \ \ \ \ \ P(R)&=P(R \lvert B_1) \times P(B_1)+P(R \lvert B_2) \times P(B_2) \\&=\frac{1}{4} \times \frac{1}{2}+\frac{2}{4} \times \frac{1}{2} \\&=\frac{3}{8} \end{aligned}

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Remark

We are not saying that an earlier event (the choosing of the box) is altered in some way by a subsequent event (the observing of a red ball). The above probabilities are subjective. How strongly do you believe that the “unknown” box is Box 1? If you use probabilities to quantify your belief, without knowing any additional information, you would say the probability that the “unknown” box being Box 1 is $\frac{1}{2}$.

Suppose you reach into the “unknown” box and get a red ball. This additional information alters your belief about the chosen box. Since Box 2 has more red balls, the fact that you observe a red ball will tell you that it is more likely that the “unknown” chosen box is Box 2. According to the above calculation, you update the probability of the chosen box being Box 1 to $\frac{1}{3}$ and the probability of it being Box 2 as $\frac{2}{3}$.

In the language of Bayesian probability theory, the initial belief of $P(B_1)=0.5$ and $P(B_2)=0.5$ is called the prior probability distribution. After a red ball is observed, the updated belief as in the probabilities $\displaystyle P(B_1 \lvert R)=\frac{1}{3}$ and $\displaystyle P(B_2 \lvert R)=\frac{2}{3}$ is called the posterior probability distribution.

As demonstrated by this example, the Bayes’ formula is for updating probabilities in light of new information. Though the updated probabilities are subjective, they are not arbitrary. We can make sense of these probabilities by assessing the long run results of the experiment objectively.

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An Insurance Perspective

The example discussed here has an insurance interpretation. Suppose an insurer has two groups of policyholders, both equal in size. One group consists of low risk insureds where the probability of experiencing a claim in a year is $\frac{1}{4}$ (i.e. the proportion of red balls in Box 1). The insureds in other group, a high risk group, have a higher probability of experiencing a claim in a year, which is $\frac{2}{4}$ (i.e. the proportion of red balls in Box 2).

Suppose someone just purchase a policy. Initially, the risk profile of this newly insured is uncertain. So the initial belief is that it is equally likely for him to be in the low risk group as in the high risk group.

Suppose that during the first policy year, the insured has incurred one claim. The observation alters our belief about this insured. With the additional information of having one claim, the probability that the insured belong to the high risk group is increased to $\frac{2}{3}$. The risk profile of this insured is altered based on new information. The insurance point of view described here has the exact same calculation as in the box-ball example and is that of using past claims experience to update future claims experience.

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Bayes’ Formula

Suppose we have a collection of mutually exclusive events $B_1, B_2, \cdots, B_n$. That is, the probabilities $P(B_i)$ sum to 1.0. Suppose $R$ is an event. Think of the events $B_i$ as “causes” that can explain the event $R$, an observed result. Given $R$ is observed, what is the probability that the cause of $R$ is $B_k$? In other words, we are interested in finding the conditional probability $P(B_k \lvert R)$.

Before we have the observed result $R$, the probabilities $P(B_i)$ are the prior probabilities of the causes. We also know the probability of observing $R$ given a particular cause (i.e. we know $P(R \lvert B_i$). The probabilities $P(R \lvert B_i)$ are “forward” conditional probabilities.

Given that we observe $R$, we are interested in knowing the “backward” probabilities $P(B_i \lvert R)$. These probabilities are called the posterior probabilities of the causes. Mathematically, the Bayes’ formula is simply an alternative way of writing the following conditional probability.

$\displaystyle (5) \ \ \ \ \ P(B_k \lvert R)=\frac{P(B_k \cap R)}{P(R)}$

In $(5)$, as in the discussion of the random experiment of choosing box and selecting ball, we are restricting ourselves to only the cases where the event $R$ is observed. Then we ask, out of all the cases where $R$ is observed, how many of these cases are caused by the event $B_k$?

The numerator of $(5)$ can be written as

$\displaystyle (6) \ \ \ \ \ P(B_k \cap R)=P(R \lvert B_k) \times P(B_k)$

The denominator of $(5)$ is obtained from applying the total law of probability.

$\displaystyle (7) \ \ \ \ \ P(R)=P(R \lvert B_1) P(B_1) + P(R \lvert B_2) P(B_2)+ \cdots + P(R \lvert B_n) P(B_n)$

Plugging $(6)$ and $(7)$ into $(5)$, we obtain a statement of the Bayes’ formula.

$\displaystyle (8) \ \ \ \ \ P(B_k \lvert R)=\frac{P(P(R \lvert B_k) \times P(B_k)}{\sum \limits_{j=1}^n P(R \lvert B_j) \times P(B_j)} \ \ \ \ \ \ \ \text{(Bayes' Formula)}$

Of course, for any computation problem involving the Bayes’ formula, it is best not to memorize the formula in $(8)$. Instead, simply apply the thought process that gives rise to the formula (e.g. the tree diagram shown above).

The Bayes’ formula has some profound philosophical implications, evidenced by the fact that it spawned a separate school of thought called Bayesian statistics. However, our discussion here is solely on its original role in finding certain backward conditional probabilities.

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Example 2

Example 2 is left as exercise. The event that both selected balls are red would give even more weight to Box 2. In other words, in the event that a red ball is selected twice in a row, we would believe that it is even more likely that the unknown box is Box 2.
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Reference

1. Feller, W., An Introduction to Probability Theory and Its Applications, third edition, John Wiley & Sons, New York, 1968.
2. Grinstead, C. M., Snell, J. L. Introduction to Probability, Online Book in PDF format.