(a) Let $X$ and $Y$ be real random variables such that $\mathbb{E}[f(X)]=\mathbb{E}[f(Y)]$ for every compactly supported continuous function $f$. Show that $X$ and $Y$ have the same law.

(b) Given a real random variable $Z$, let $\varphi_{Z}(s)=\mathbb{E}\left(e^{i s Z}\right)$ be its characteristic function. Prove the identity

$$\iint g(\varepsilon s) f(x) e^{-i s x} \varphi_{Z}(s) d s d x=\int \hat{g}(t) \mathbb{E}[f(Z-\varepsilon t)] d t$$

for real $\varepsilon>0$, where is $f$ is continuous and compactly supported, and where $g$ is a Lebesgue integrable function such that $\hat{g}$ is also Lebesgue integrable, where

$$\hat{g}(t)=\int g(x) e^{i t x} d x$$

is its Fourier transform. Use the above identity to derive a formula for $\mathbb{E}[f(Z)]$ in terms of $\varphi_{Z}$, and recover the fact that $\varphi_{Z}$ determines the law of $Z$ uniquely.

(c) Let $X$ and $Y$ be bounded random variables such that $\mathbb{E}\left(X^{n}\right)=\mathbb{E}\left(Y^{n}\right)$ for every positive integer $n$. Show that $X$ and $Y$ have the same law.

(d) The Laplace transform $\psi_{Z}(s)$ of a non-negative random variable $Z$ is defined by the formula

$$\psi_{Z}(s)=\mathbb{E}\left(e^{-s Z}\right)$$

for $s \geqslant 0$. Let $X$ and $Y$ be (possibly unbounded) non-negative random variables such that $\psi_{X}(s)=\psi_{Y}(s)$ for all $s \geqslant 0$. Show that $X$ and $Y$ have the same law.

(e) Let

$$f(x ; k)=1_{\{x>0\}} \frac{1}{k !} x^{k} e^{-x}$$

where $k$ is a non-negative integer and $1_{\{x>0\}}$ is the indicator function of the interval $(0,+\infty)$.

Given non-negative integers $k_{1}, \ldots, k_{n}$, suppose that the random variables $X_{1}, \ldots, X_{n}$ are independent with $X_{i}$ having density function $f\left(\cdot ; k_{i}\right)$. Find the density of the random variable $X_{1}+\cdots+X_{n}$.