Skip to content

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.

2 Comments Post a comment
  1. Josh G #

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

    April 20, 2014
  2. Scott Alister McKinley #

    That’s correct, Josh. Good catch.

    April 21, 2014

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s