Friday 6 March 2015

Examples Sheet 3

Most of you will have now done Examples Sheet 3. Here are a few notes.

 #4. You'll want to compare this with Examples sheet 4 #5.

 #5. What would be the analogous question to this one, posed for continuous r.vs?

 #8. You should know how to answer this two ways: by using p.g.fs, and also directly via that tower property of conditional expectation.

 #16. How do we know that the answer is $a_z=a^z$, where $a$ the smallest positive root of $a=pa^3+1-p$? We have such a theorem for the extinction probability in a branching processes. How can this coin tossing problem be viewed through the lens of some branching process?

#17. If you need a hint, here is one: $P(|X+Y|>\epsilon)\leq P(|X|\geq \epsilon/2)+P(|Y|\geq \epsilon/2)$.

 #18. The hint is for the third part. But it also useful in answering the second part: "Could you load the die so that the totals {2,3,4,5,6,7} are obtained with equal probabilities?"