Paper 2, Section I, 4 F4 \mathrm{~F}

Probability | Part IA, 2016

Define the moment-generating function mZm_{Z} of a random variable ZZ. Let X1,,XnX_{1}, \ldots, X_{n} be independent and identically distributed random variables with distribution N(0,1)\mathcal{N}(0,1), and let Z=X12++Xn2Z=X_{1}^{2}+\cdots+X_{n}^{2}. For θ<1/2\theta<1 / 2, show that

mZ(θ)=(12θ)n/2.m_{Z}(\theta)=(1-2 \theta)^{-n / 2} .

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