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 [1].

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

and

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.

**Reference**

- Feller W.,
*An Introduction to Probability Theory and Its Applications, Vol II, Second Edition*, John Wiley & Sons (1971)

Thanks, I knew the t-distribution was a mixture of normals, but I was using slightly the wrong formula to generate a t-distribution, and this set out what I needed to do very clearly. FWIW, I have found that a t-distribtion with >2 d.f. can be emulated extremely closely with just 4 suitably chosen and weighted (by a formula) normal distributions. I needed to do this as I wanted to add the square of a t-distributed variable with a low d.f. to a chi-squared varaible (actually a F distributed varaibel, with the second F d.f. large)