Paper 1, Section II, F

Probability | Part IA, 2020

(a) Let ZZ be a N(0,1)N(0,1) random variable. Write down the probability density function (pdf) of ZZ, and verify that it is indeed a pdf. Find the moment generating function (mgf) mZ(θ)=E(eθZ)m_{Z}(\theta)=\mathbb{E}\left(e^{\theta Z}\right) of ZZ and hence, or otherwise, verify that ZZ has mean 0 and variance 1 .

(b) Let (Xn)n1\left(X_{n}\right)_{n \geqslant 1} be a sequence of IID N(0,1)N(0,1) random variables. Let Sn=i=1nXiS_{n}=\sum_{i=1}^{n} X_{i} and let Un=Sn/nU_{n}=S_{n} / \sqrt{n}. Find the distribution of UnU_{n}.

(c) Let Yn=Xn2Y_{n}=X_{n}^{2}. Find the mean μ\mu and variance σ2\sigma^{2} of Y1Y_{1}. Let Tn=i=1nYiT_{n}=\sum_{i=1}^{n} Y_{i} and let Vn=(Tnnμ)/σnV_{n}=\left(T_{n}-n \mu\right) / \sigma \sqrt{n}.

If (Wn)n1\left(W_{n}\right)_{n \geqslant 1} is a sequence of random variables and WW is a random variable, what does it mean to say that WnWW_{n} \rightarrow W in distribution? State carefully the continuity theorem and use it to show that VnZV_{n} \rightarrow Z in distribution.

[You may not assume the central limit theorem.]

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