** Problems: Assignment 2**

Suppose **X** and **Y** have joint density . Prove from the
definition of density that the density of **X** is .

Suppose **X** is Poisson(). After observing **X** a coin landing
Heads with probability **p** is tossed **X** times. Let **Y** be the number of
Heads and **Z** be the number of Tails. Find the joint and marginal distributions
of **Y** and **Z**.

Let be the bivariate normal density with mean 0, unit variances and correlation and let be the standard bivariate normal density. Let .

- Show that
**p**has normal margins but is not bivariate normal. - Generalize the construction to show that there rv's
**X**and**Y**such that**X**and**Y**are each standard normal,**X**and**Y**are uncorrelated but**X**and**Y**are not independent.

** Warning: This is probably hard. Don't waste too much time on
it.** Suppose **X** and **Y** are independent and
random variables. Show that is a random variable.

Suppose **X** and **Y** are independent with
and . Let **Z=X+Y**.
Find the distribution of **Z** given **X** and that of **X** given **Z**.

Thu Oct 10 22:10:13 PDT 1996