Probability Theory, Homework 3, due Friday Oct 3.
From Jacod and Protter
9.5 Let 
9.7 Suppose that 
Further exercises
Exercise 3. Suppose that 
- Prove
.
- If
, the factor
may be replaced with
.
- If
, the factor
Exercise 4. Suppose that 
CORRECTION: 
Hint: Think Cauchy-Schwarz.
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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.