Stochastics: HW 7, due Wed Apr 23

Scott Alister McKinleyApril 18, 2014Stochastics Homework

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 $\mathcal{F}_n$ .

2 thoughts on “Stochastics: HW 7, due Wed Apr 23”

Josh G says:

April 20, 2014 at 5:10 pm

In problem 2, are X and Y independent (for E[Z|X])?

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Scott Alister McKinley says:

April 21, 2014 at 8:19 am

That’s correct, Josh. Good catch.

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