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Mean and variance of chi-squared random variables (2019-01-29)

The chi-squared distribution ${\chi }_k^2$ with degree of freedom $k$ is defined as the following random variable $Q$'s distribution: $$Q=\sum _{i=1}^k{X}_i^2$$ where $\{X_i\}(i=1,2,...,k)$ is a sequence of independent standard normal random variables.

It is simple to prove that (1) $Q$'s mean is $k$ and (2) $Q$'s variance is $2k$; for (1), $$\begin{array}{rl}\mathbb{E}[Q]& =\mathbb{E}[\sum _{i=1}^k{X}_i^2]\\ & =\sum _{i=1}^k\mathbb{E}[X_i^2]\text{(linearity of expectation)}\\ & =\sum _{i=1}^k\mathrm{Var}(X_i)(\text{as }\mathbb{E}[X_i]=0)\\ & =\sum _{i=1}^{k}1\\ & =k.\end{array}$$

For (2), $$\begin{array}{rl}\mathrm{Var}(Q)& =\mathrm{Var}(\sum _{i=1}^k{X}_i^2)\\ & =\sum _{i=1}^k\mathrm{Var}(X_i^2)(\text{as }{X}_i\text{s are independent})\\ & =\sum _{i=1}^k(\mathbb{E}[X_i^4]-\mathbb{E}[X_i^2{]}^2)\\ & =\sum _{i=1}^k(\mathbb{E}[X_i^4]-1)(\text{as }\mathbb{E}[X_i^2]=\mathrm{Var}(X_i)=1)\\ & =\sum _{i=1}^k(3-1)(\text{as }\mathbb{E}[X_i^4]=\mathrm{Kurt}(X_i)=3)\\ & =2k\end{array}$$ where $\mathrm{Kurt}(X_i)$ is the kurtosis of $X_i$.