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

The chi-squared distribution χk2 with degree of freedom k is defined as the following random variable Q's distribution: Q=i=1kXi2 where {Xi}(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), E[Q]=E[i=1kXi2]=i=1kE[Xi2](linearity of expectation)=i=1kVar(Xi)(as E[Xi]=0)=i=1k1=k.

For (2), Var(Q)=Var(i=1kXi2)=i=1kVar(Xi2)(as Xis are independent)=i=1k(E[Xi4]E[Xi2]2)=i=1k(E[Xi4]1)(as E[Xi2]=Var(Xi)=1)=i=1k(31)(as E[Xi4]=Kurt(Xi)=3)=2k where Kurt(Xi) is the kurtosis of Xi.


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