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# Homework Assignments

## Homework 3

1. Suppose and are the indicator functions of events and , respectively. Find , and show that and are independent if and only if .
2. Suppose is a ch.f. Show that is also a ch.f.
3. Suppose and are jointly Gaussian. Use ch.f.s to show that .
4. Suppose and are jointly continuous. (a) Show that

and thus that

(b) Suppose . Show that
5. Suppose and are independent continuous r.v.s with c.d.f.s and , respectively. Suppose further that for all . Show that
6. Prove Jensen's inequality for the case of simple-function r.v.'s
7. Prove the Schwartz inequality.

Problems from Grimmet & Stirzaker :

1. Prob 2.7.4
2. Prob 2.7.7. Hint: binomial distribution
3. Prob 2.7.9. Hint:

4. Ex 3.3.1. Hint: Let , and .
5. Ex 3.4.1. Let be the indicator function of the event that the outcome of the st toss is different from the outcome of the th toss. The number of distinct runs is . Observe that and are independent if . Show that

Show that
6. Ex. 3.4.2. Hint: Let , where is the number on the th ball. Show that: . show that . Hint:

7. Ex 3.5.2. Hint: .
8. Ex 3.6.5. Hint: , with equality if and only if .