(i) Define the moment generating function $M_{X}(t)$ of a random variable $X$. If $X, Y$ are independent and $a, b \in \mathbb{R}$, show that the moment generating function of $Z=a X+b Y$ is $M_{X}(a t) M_{Y}(b t)$.

(ii) Assume $T>0$, and $M_{X}(t)<\infty$ for $|t|<T$. Explain the expansion

$$M_{X}(t)=1+\mu t+\frac{1}{2} s^{2} t^{2}+\mathrm{o}\left(t^{2}\right)$$

where $\mu=E(X)$ and $s^{2}=E\left(X^{2}\right) . \quad$ [You may assume the validity of interchanging expectation and differentiation.]

(iii) Let $X, Y$ be independent, identically distributed random variables with mean 0 and variance 1 , and assume their moment generating function $M$ satisfies the condition of part (ii) with $T=\infty$.

Suppose that $X+Y$ and $X-Y$ are independent. Show that $M(2 t)=M(t)^{3} M(-t)$, and deduce that $\psi(t)=M(t) / M(-t)$ satisfies $\psi(t)=\psi(t / 2)^{2}$.

Show that $\psi(h)=1+\mathrm{o}\left(h^{2}\right)$ as $h \rightarrow 0$, and deduce that $\psi(t)=1$ for all $t$.

Show that $X$ and $Y$ are normally distributed.