Sun 6 September 2020

The t-distribution becomes Normal as n increases

Written by Hongjinn Park in Articles

Why does the t-distribution become Normal as $n \rightarrow \infty$?

First let's construct a $t_n$ random variable from Normals. If $Z, Z_1, ..., Z_n$ are independent and identically distributed standard Normal random variables then the following random variable,

$$t_n = \frac{Z}{\sqrt{\frac{Z_1^2+...+Z_n^2}{n}}}$$

has a t-distribution with $n$ degrees of freedom.

Now as $n \rightarrow \infty$, $t_n$ becomes Normal. We can see this intuitively by using the Law of Large Numbers.

First note that the expected value of a Chi Squared RV is one. In other words, $E[Z^2] = 1$. The fraction in the square root is the sample mean of $n$ iid $Z_i^2$ RVs. By the LLN the sample mean as $n$ gets large will equal the expected value of a Chi Squared RV which is one.

So as $n$ gets really big you're just left with $\frac{Z}{\sqrt{1}} = Z$ which shows how $t_n$ becomes a Normal random variable.



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