Student's T-Distribution

Topics: Probability - Standard Normal Distribution - Chi-Squared Distribution

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(definition)

Let $X$ and $Y$ be independent random variable such that $X\sim \mathcal{N}(0,1)$ and $Y\sim {\chi }_{(n)}^2$. Let:

$$W=\frac{X}{\sqrt{\frac{Y}{n}}}$$

In such a case, we say that $W$ has a Student’s t-distribution with $n$ degrees of freedom.

Expected Value and Variance §

(theorem)

A random variable $W$ with a Student’s t-distribution has an expected value and variance of:

$$\mathbb{E}[W]=0$$

$$Var(W)=\frac{n}{n-2}$$