Suppose $M_{n}$ is a non-negative Martingale. Use Theorem $5.13$ to show that for all $\lambda>0$
$$
\mathbb{P}\left(\max {0 \leq k \leq n} M{k}>\lambda\right) \leq \frac{\mathbb{E}\left(M_{n}\right)}{\lambda}
$$
Note that the above inequality is an improvement of the well-known Markov’s inequality
$$
\mathbb{P}(|X|>\lambda) \leq \frac{\mathbb{E}|X|}{\lambda}
$$

Suppose $M_{n}=X_{1}+\cdots+X_{n}$ with $M_{0}=0$ is a Martingale wrt $X_{n}$. Show that (a) for all $n \geq 1, \mathbb{E}\left(X_{n}\right)=0$.
(b) $\operatorname{Var}\left(M_{n}\right)=\operatorname{Var}\left(X_{1}\right)+\cdots+\operatorname{Var}\left(X_{n}\right)$

Consider a discrete M.C. on $S={0,1,2, \ldots}$ whose transition probabilities are defined as follows: $p(k, \ell)=\mathbb{P}{Y+Z=\ell}$ where $Y \sim \operatorname{binom}(k, p)$ and $Z \sim \operatorname{Pois}(\lambda)$ are independent. Suppose further that $X_{0} \sim \operatorname{Pois}(\gamma)$
(a) What is the distribution of $X_{1}$ ?
(b) What is the distribution of $X_{n}$ ?
(c) Find the limiting distribution of $X_{n}$ as $n \rightarrow \infty$, i.e. $\lim {n \rightarrow \infty} \mathbb{P}\left(X{n}=k\right)$.
(d) Find the stationary distribution of the chain.
HINT: moment-gencrating function nethod.