# Probability Theory, Homework 3, due Friday Oct 3.

**From Jacod and Protter**

9.5 Let be a probability space. Suppose that is a random variable with almost surely and . Define by . Show that defines a probability measure on .

9.7 Suppose that and let be defined as above. Let denote expectation with respect to . Show that .

**Further exercises**

Exercise 3. Suppose that and are random variables and that .

- Prove .
- If , the factor may be replaced with .
- If , the factor can be replaced with 1.

Exercise 4. Suppose that and . Prove for that

CORRECTION: .

Hint: Think Cauchy-Schwarz.

Prob 4. Let a=1/sqrt(1.81). Put X =0.1a wp 1/2 and 1.9a wp 1/2. Then E(X^2)=1 and E(X) = a.

Let lamda = 0.11, then

P(X>= 0.11a) = 1/2 and

(1-0.11^2)/1.81=0.54 >P(X>= 0.11a).

For the last problem, exercise 2, should the conclusion read P( |X| < \lambda *a) instead of P( |X| \geq \lambda * a)?

Should the last problem conclusion read “P ( |X| < \lambda *a)" rather than "P ( |X| \geq \lambda * a)?

I think I've shown the former, the later seems much harder, although I haven't given up trying, just thought I'd double check.

Appears by Hung’s Counterexample (If I’m reading it right) that we did want the inequality inside the 4th exercise to be < instead of \geq… unless someone has anything else to add?

Hi guys, Looks like you’re right. I’ve made the adjustment to the statement of the problem. Sorry for the typo.

Nope. Everyone was wrong. I think I’ve got a proof of the corrected statement above for problem 4.

Should 3(c) say the factor 2^p can be replaced with 1?

Yes. That’s right. Sorry for the typo.