Most of you will have now done Examples Sheet 3. Here are a few notes.
#3. I have amended this question to read "Prove that for $t > 0$, ...." Why?
#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 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?
#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?"
#3. I have amended this question to read "Prove that for $t > 0$, ...." Why?
#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 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?
#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?"
