# Paper 1, Section II, F

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

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

(c) Let $Y_{n}=X_{n}^{2}$. Find the mean $\mu$ and variance $\sigma^{2}$ of $Y_{1}$. Let $T_{n}=\sum_{i=1}^{n} Y_{i}$ and let $V_{n}=\left(T_{n}-n \mu\right) / \sigma \sqrt{n}$.

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

[You may not assume the central limit theorem.]