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STAT 801

Problems: Assignment 3

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

  1. Find the joint density of .

  2. Suppose . Prove that is independent of .

  3. Find the density of .

  4. Find the density of .

Suppose are iid exponential. Let .

  1. Find the joint density of .

  2. Find the joint density of .

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

  1. Develop a recurrence relation for and , expressing and in terms of and .

  2. Find the joint density of .

  3. 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 .

  1. Show that and are independent.

  2. Show that is uniformly distributed on .

  3. Show is a Cauchy random variable.





Richard Lockhart
Thu Oct 10 22:15:37 PDT 1996