Where does “Student’s t-distribution” come from? Well, suppose that is a standard normal random variable and is a chi-squared random variable with degrees of freedom. In addition, suppose that and are independent. Define a random variable by
We call the distribution of the quotient random variable the -distribution with degrees of freedom. Its distribution is absolutely continuous with respect to the Lebesgue measure, and we calculate its probability density function (pdf) now. Recall that has pdf
Using the calculus, we obtain the pdf of :
for all . Since and are independent, their joint density function is simply the product of the marginal density functions. Hence,
Making the change of variable , we conclude that
Statistics textbooks–particularly, bad ones–state that the normal distribution approximates the t-distribution for sufficiently large degrees of freedom; the quantity behind sufficiently varies from author to author. I’m more interested in mathematical proofs than taking an author’s word, so let’s examine for ourselves the limiting behavior of the t-distribution as .
Using the below version of Stirling’s formula,
we see that
where we use the sequence limit definition of and the continuity of the exponential function.
Since as , we see that, for any ,
Letting , we see that
The preceding inequality holds for arbitrarily small , so we conclude that
Putting these results together, for fixed ,
where the right-hand side is the density function of the standard normal distribution.