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Homework 2

1. Suppose is a r.v. with c.d.f. . Prove the following:
1. is nondecreasing.
2. .
3. .
4. is right continuous.
5. if .
Also, find expressions for , and in terms of .
2. Show that the following are valid p.m.f.s:
1. Binomial: if .
2. Poisson: for .
3. Find the mean and variance of when is (a) ; (b) Binomial ; (c) Poisson ; (d) Exponential . Do not use characteristic functions.
4. Suppose that and are jointly continuous. Show that

5. Suppose that and are jointly Gaussian with parameters . Show that .
6. Suppose , and define . Are and uncorrelelated? Are and independent? Find the p.d.f. of . Are and jointly continuous?
7. Show that .
8. Suppose . Use the ch.f. of to find an expression for , .

Problems from the Grimmett & Stirzaker text:

1. Ex 1.4.4
2. Ex 1.4.5. Hint: Let be the event that the th door conceals the car, let be the event that you see a goat, and let be the event that you see Bill.
3. Ex 1.5.1.
4. Ex 1.5.2
5. Ex 1.5.7.
6. Prob. 1.8.5
7. Prob. 1.8.6.
8. Prob. 1.8.19.
9. Prob. 1.8.20.
10. Prob. 1.8.30.