** Problems: Assignment 3**

Suppose are iid real random variables with
density ** f ** . Let be the ** X **'s arranged
in increasing order.

- Find the joint density of .
- Suppose . Prove that is independent of .
- Find the density of .
- Find the density of .

Suppose are iid exponential. Let .

- Find the joint density of .
- Find the joint density of .

Suppose are iid N(,). Let . Let .

- Develop a recurrence relation for and , expressing
and in terms of and .
- Find the joint density of .
- Generate data from N(0,1). By adding to the data for
some large values of
**k**compare the numerical performance of these recurrence relations to that of the one pass formula using , and the usual computing formulas for the sample variance.

Suppose **X** and **Y** are iid .

- Show that and are independent.
- Show that is uniformly
distributed on .
- Show is a Cauchy random variable.

Thu Oct 10 22:15:37 PDT 1996