The Stuent’s t distribution is an important distribution in statistics. It is the basis for the Student’s t statistic and arises in the problem of etimating the mean of a normal population. The Student’s t distribution (t distribution for short) is usually defined as the following ratio
where and has a chi-square distribution with degrees of freedom. For this derivation, see .
In this post we discuss another way of deriving the t distribution. The alternative view is through the notion of mixture (compounding in some texts). Suppose that has a normal distribution with mean and variance . That is, . There is uncertainty in the variance . Further suppose that follows a gamma distribution with parameters and where and is a positive integer. Then the unconditional distribution of has a Student’s t distribution with degrees of freedom. In the language of mixture in probability, we say that the Student’s t distribution is a mixture of normal distributions with gamma mixing weights.
The conditional distribution of is where
The marginal (the unconditional) density function of is obtained by integrating out the parameter . The resulted density is that of the Student’s t distribution. The following is the derivation.
Now, let . Then the above density function becomes:
The above density function is that of a Student’s t distribution with degrees of freedom. It is interesting to note that because of the uncertainty in the parameter , the Student’s t distribution has a longer tail than the conditional normal distribution used in the beginning of the derivation.
- Feller W., An Introduction to Probability Theory and Its Applications, Vol II, Second Edition, John Wiley & Sons (1971)