📂Probability Distribution

The Mean and Variance of the Chi-Squared Distribution

Formula

If $X \sim \chi^{2} (r)$ then $$E(X) = r \\ \Var (X) = 2r$$

Derivation

Strategy: Fortunately, the moment generating function of the chi-squared distribution is known.

Moment of the chi-squared distribution: Let’s say $X \sim \chi^{2} (r)$. If $k > - r/ 2$, then there exists the $k$th moment $$E X^{k} = {{ 2^{k} \Gamma (r/2 + k) } \over { \Gamma (r/2) }}$$

Mean

$$EX^{1} = {{ 2^{1} \Gamma (r/2 + 1) } \over { \Gamma (r/2) }} = 2 \cdot {{ r } \over { 2 }} = r$$

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Variance

$$EX^{2} = {{ 2^{2} \Gamma (r/2 + 2) } \over { \Gamma (r/2) }} = 4 \cdot {{ r + 2 } \over { 2 }} \cdot {{ r } \over { 2 }} = r^{2} + 2r$$ Therefore, $$\Var(X) = \left( r^{2} + 2r \right) - r^{2} = 2r$$

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