# Stochastics: HW 7, due Wed Apr 23

1. The Autocovariance problem posted last week..

2. Consider the random variable

$\displaystyle X := \left\{ \begin{matrix} 5 & \text{wp } 1/2 \\ 2 & \text{wp } 1/3 \\ 1 & \text{wp } 1/6 \end{matrix} \right.$

and let $A$ be the event that $X = 2$ or $X = 5$.

• Compute $\mathbb{E}[X \, | \, A]$.
• Compute $\mathbb{E}[X 1_{A}]$.
• Let $Y$ be distributed like $X$ and let $Z := X + Y$.  Compute $\mathbb{E}[Z \, | \, A]$.
• Compute $\mathbb{E}[Z \, | \, X]$.

3. Let $\{X_1, \, X_2, \, \ldots\}$ be a sequence of iid random variables and let $\mathcal{F}_n = \sigma(X_1, X_2, \ldots, X_n)$.  Let $m(s) = \mathbb{E}[e^{sX_1}]$ be the moment generating function of $X_1$ (and hence of each $X_i$).  Fix $s$ and assume $m(s) < \infty$.  Let $S_0 = 0$ and for $n > 0$,

$S_n = X_1 + X_2 + \ldots + X_n$.

Define $M_n = m(s)^{-n} e^{s S_n}$. Show that $M_n$ is a martingale with respect to $\mathcal{F}_n$.