Discussion
The variance of a Chi-square distribution with n degrees of freedom is :
Answer: A
The variance is \(2n\).
A chi-square variable with \(n\) degrees of freedom is the sum of the squares of \(n\) independent standard normal variables. Each \(Z^{2}\) has mean 1 and variance 2, because \(E(Z^{4}) = 3\):
\[ \begin{aligned} \operatorname{Var}(Z^{2}) &= E(Z^{4}) - [E(Z^{2})]^{2} \\ &= 3 - 1 = 2 \\ \operatorname{Var}(\chi^{2}_{n}) &= n \times 2 = 2n \end{aligned} \]
\(n\) is the mean of the distribution, not its variance, which is the usual trap. \(n^{2}\) and \(\sqrt{n}\) do not arise for chi-square; \(\sqrt{2n}\) would be its standard deviation.
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