Chi-Squared Distribution

Topics: Probability - Standard Normal Distribution

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(definition)

Let $Z_1,\dots ,Z_k$ be random variables with a standard normal distribution. Let:

$$X=Z_1^2+\cdots +Z_k^2$$

In such a case, we say that $X$ follows a chi-squared distribution with $k$ degrees of freedom and write $X\sim {\chi }_{(k)}^2$.

Expected Value and Variance §

(theorem)

A random variable $X$ such that $X\sim {\chi }_{(k)}^2$ has an expected value and variance of:

$$\mathbb{E}[X]=k$$

$$\mathrm{Var}(X)=2k$$