Welcome to Math 5310 (Probability)!
Homework will (usually) be posted here on Mondays or Tuesdays, and due in class the following
Monday (unless stated otherwise). We will usually go over the homeworks when they are due, sometimes followed by a quick quiz
on the material.
Chapter 1 problems #2, 6, 7, 10, 14, 17, 21, 27, 42
(1) Flip a fair coin 4 times and let X be the number of Heads. Then flip the coin X times and let Y be the number of Heads.
(a) Find P(Y=k), k=0,1,2,3,4
(b) Find P(X=i | Y=j), i,j = 0,1,2,3,4
(2) Coin#1 has P(H) = p, Coin#2 has P(H) = q. Flip each coin twice. Let X be the number of Heads from Coin#1, and let Y be the number of Heads from Coin#2.
Find P(X=n | X+Y=m) for n=0,1,2 and m = 0,1,2,3,4
(3) Find the probability that a hand with six cards has "3 pairs".
Chapter 1, problems #46,47,48,49,50,60,61
(most of these have answers in the text, so read them over and try to understand.)
Chapter 2, problems #2, 3, 7a, 11, 16, 18, 21, 23
(1) Roll 2 dice (X1,X2), and let X = X1 + X2 (sum of the 2 dice); then flip a fair coin X times, and let Y be the number of Heads.
(a) Find the probability mass function for Y (Hint: Condition on X)
(b) Find E(Y) and Var(Y) (Hint: condition on X)
(c) Repeat (a) and (b) when P(Heads) = p.
(2) Roll 2 dice (X1,X2), and let Y = min(X1,X2), Z = max(X1,X2).
(a) Find E(Y | Z=i) and Var(Y | Z=i), i=1,2,3,4,5,6.
(b) Use part (a) to find E(Y) by conditioning on Z. It should check the answer we got in class as few weeks ago.
Text, chapter 2, #39,40,41
Here are Midterm Solutions
Sorry foor the delay - I was traveling.
Text, chapter 3, #7, 8, 21
Let X, Y be random variables with joint density f(x,y) = c(x^2 + 2y^2), x,y in [0,1].
(a) Find c.
(b) Find the marginal densities for X and Y.
(c) Find E(X), E(Y), E(XY).
(d) Find the conditional density of X given that Y = y.
Text, Chapter 3, #34
Text, Chapter 4, #14, 17, 18, 19, 22
The due date for Homework #7 will be wednesday 10/18, instead of monday, with the exception of #34 from Chapter 3. Try to work that one out before class monday. The other problems use techniques we will talk about on monday. Feel free to read ahead in chapter 4!
The second midterm exam will be postponed one week. It will be on Wednesday November 1, and cover through chapter 4.
Text, Chapter 4, #25, 26 (answers follow the problems)
More practice on conditioning:
(1) Let X ~ expon(lambda). Then flip a coin that come up Heads with probability exp(-X).
(a) Find P(H)
(b) Find the conditional density of X given the flip is Heads.
(c) Find the conditional density of X given the flip is Tails.
(d) Find E(X|H) and Var(X|H).
(e) Now suppose you flip the coin until you get heads. Let Y be the number of flips needed.
Find the conditional density of X given Y=k.
Text, Chapter 4, #1,2,3 (I didn't assign any of these kinds of problems yet!)
Text, Chapter 4, #30, 42, 45 (problems on moment generating functions)
1) Let N_1(t) and N_2(t) be independent Poisson processes with rates \lambda_1 and \lambda_2. Let N(t) = N_1(t) + N_2(t). Let T_1(i) and T_2(i) be the times of the i-th event for Poisson processes 1 and 2. Find
(a) P(N(t) = k)
(b) P(N(t) = n | N_1(t) = k), k < or = n
(c) P(N_1(t) = k | N(t) = n), k > or = n
(d) P(T_1(1) < T_2(1))
(e) P(T_1(2) < T_2(1))
(f) E(N(t) | T_1(1) = s), s < t
2) Phone calls occur at a call center as a Poisson process with rate 10 per hour. You have to work at the call center until you answer 10 phone calls or 1 hour passes (which ever happens first). Let N(t) be the number of calls up to time t, and let T be the amount of time (in hours) that you work. Find experssions for the following (don't try to simplify too much)
(a) P(T = 1), P(N(.5)=k | T=1)
(b) E(T), E(N(.5) | T=1)
3) Recall in class we found P(T2 < s | N(t)=2)=(s/t)^2 by conditioning on T2. See if you can get the same answer by conditioning on T1.
1) Recall the Urn game example from class. This time there are 4 Red balls and 4 blue balls, with 4 in each Urn. Each turn, you take a ball out of Urn#1 and put it in Urn#2, then take a ball from Urn#2 and put it in Urn #1.
a) Find the transition matrix for the Markov chain. (The states are {0,1,2,3,4})
b) Find the steady state vector, "pi".
c) Find the mean and variance of the number of transitions to get from state 0 to state 4. (You can use a computer.)
d) Starting from state 1, what is the probability you reach state 0 before you reach state 4?
2) There are two coins, the first has P(Heads) = p_1, the second has P(Heads) = p_2. You choose a coin at random and flip it repeatedly. Let X_n = 1 if the n-th flip is heads and X_n = 0 if it is Tails. Is {X_1, X_2, ...} a Markov chain? Hint: Find P(X_3 = 1 | X_2 = 1, X_1 = 1) and P(X_3 = 1 | X_2 = 1)