Math 318 Assignment 2:
Due Monday, January 23 at start of class
1. A government wants to nd out how many of its citizens are lling out fraudulent
tax returns. It creates a survey with the single question Do you ll out your tax
return honestly? Along with
Math 130B, Homework #5
Due: February 17, in class
Some review
Problem 1.
1. In class I showed that E(X) =
random variables
n>0
P (X > n) by just writing out the sum. Instead, lets dene the
In =
if X n
if X < n
1
0
and express X in terms of the In s. Use t
Math 318 Assignment 3:
Due Monday, January 30 at start of class
1. Two teams A and B play a series of games, until one team wins four games.
Suppose team A has probability p of winning each game, and teanm B has probability
1 p.
(a) Compute the probabilit
Math 318 Assignment 2 solutions
1. A government wants to nd out how many of its citizens are
lling out fraudulent tax returns. . .
Let F denote the event that a taxpayer ll out a fraudulent return, so F c is the event
that a taxpayer ll out an honest retu
Math 318 Assignment 4:
Due Monday, February 6 at start of class
1. Chapter 2 #42 (Suppose that every day you receive a coupon in the mail.)
2. Chapter 5 #4 (Assume that all service times are independent.)
3. A random variable X has exponential distributio
Math 318 Assignment 5:
Due Monday, February 13 at start of class
1. A contestant in a game show gets to pick at random one of two envelopes. One has
amount X in it, and the other 2X. After looking at his prize, he is given one chance
to switch. Suppose X
Math 318 Assignment 3 solutions
1. Two teams A and B play a series of games, until one team wins four games. Suppose
team A has probability p of winning each game, and team B has probability 1 p.
(a) Compute the probability that team A wins the series, as
Math 318 Assignment 5:
Due Monday, February 13 at start of class
1. A contestant in a game show gets to pick at random one of two envelopes. One has
amount X in it, and the other 2X. After looking at his prize, he is given one chance to
1
switch. Suppose
Math 318 Assignment 6:
Due Wednesday, February 29 at start of class
1. Chapter 5 #39. For parts (c,d,e), use the central limit theorem.
2. ESP cards have one of ve shapes on them (cross, circle, square, star and waves) with
equal probability. A psychic cl
Math 318 Assignment 4:
Due Monday, February 6 at start of class
1. Chapter 2 #42 (Suppose that every day you receive a coupon in the mail.)
Solution: Let Xi be the number of coupons needed to get the ith new type, once
we already have i 1 dierent types. T
Math 318 Assignment 6:
Due Wednesday, February 29 at start of class
1. Chapter 5 #39. For parts (c,d,e), use the central limit theorem.
Solution:
(a) The intervals Xi between mistakes are Exp(2.5) years, so they each have expec1
tation 2.5 . Therefore the
Math 318 Assignment 1:
Due Monday, January 16 at start of class
Note: In all solutions involving permutations and combinations, be sure to briey explain all
factors arising in your solution.
1. Ross: Chapter 1, #3.
2. Ross: Chapter 1, #4.
3. The following
Math 318 Assignment 1 solutions
1. Ross: Chapter 1, #3.
The sample space is sequences of hheads (H) and tails (T) that terminate withg two
heads but with no earlier pair of consecutive heads. This is an innite sample space.
The shorter entries are cfw_HH,
Solutions for Homework 2.
Problem 1. Write S for the event the message is spam, and N S for the even that its not spam. By Bayes
Theorem,
P (W1 W2 |S)P (S)
P (W1 W2 |S)P (S) + P (W1 W2 |N S)P (N S)
P (W1 |S)P (W2 |S)P (S)
=
P (W1 |S)P (W2 |S)P (S) + P (W1
Math 130B, Homework #2
Due: January 25, in class
Bayes Theorem
Problem 1. Recall the Bayesian Spam Filter from class, where W1 is the event of seeing the word viagra
in a given message, and W2 is the event of seeing the wordenhancement (for example). Now
Solutions for Homework 3.
Problem 1.
1. Notice that 0 X Y always. Furthermore, by the Binomial theorem we have
k
2k = (1 + 1)k =
j=0
k
.
j
Hence
k
P (Y = k) =
k e2 k
k!
j
P (X = j, Y = k) =
j
j=0
e2 k
e2 (2)k
=2k
=
.
k!
k!
That is, we see that Y has the P
Math 130B, Homework #4
Due: February 10, in class
Problem 1. The joint p.m.f. pX,Y (a, b) of the random variables X and Y is given by
p(1, 1) = 1 ,
9
p(1, 2) = 1 ,
9
p(1, 3) = 0,
p(2, 1) = 1 ,
3
p(2, 2) = 0,
p(2, 3) = 1 ,
6
p(3, 1) = 1 ,
9
1
p(3, 2) = 1
Math 130B, Homework #3
Due: February 3, in class.
A Joint distribution
Problem 1. Suppose that the joint p.m.f. for the random variables X and Y is
P (X = j, Y = k) =
k e2 k
,
j
k!
0 j k.
is just a positive parameter.
1. Find the p.m.f. of Y . (Recall th
Math 130B, Homework #4
Due: February 10, in class
Problem 1. The joint p.m.f. pX,Y (a, b) of the random variables X and Y is given by
p(1, 1) = 1 ,
9
p(1, 2) = 1 ,
9
p(1, 3) = 0,
p(2, 1) = 1 ,
3
p(2, 2) = 0,
p(2, 3) = 1 ,
6
p(3, 1) = 1 ,
9
1
p(3, 2) = 1
Solutions for Homework 5.
Problem 1.
1. Notice that X = n if and only if I1 = I2 = In = 1 and Ik = 0 for k > n. Hence
X=
Ij ,
j1
so taking expectations of both sides gives
E(X) =
P (X j).
E(Ij ) =
j1
j1
2. Using the same In s as above, dene the similar Jm
Math 130B, Homework #7
Due: March 9, in class
Problem 1. For a two-state cfw_0, 1 Markov Chain with general transition probabilities
1p
q
p
1q
Derive:
1. The conditions on p and q for the chain to be irreducible
2. The conditions on p and q for the chain
Notes on Joint Distributions for Math 130B
Joint probability mass functions
Suppose you run an experiment, and want to keep track of two features at the same time. For instance, lets
roll a 6-sided die two times, and keep track of X = #of 5 s, and Y = #of
1
Gamblers Ruin Problem
Consider a gambler who starts with an initial fortune of $1 and then on each successive gamble
either wins $1 or loses $1 independent of the past with probabilities p and q = 1 p respectively.
Let Rn denote the total fortune after
Math 130B, Homework #6
Due: March 2, in class
Problem 1. Let Y0 , Y1 , Y2 , . . . be independent Bernoulli(1/2) random variables. For n 1 let
Xn = Yn + Yn1 .
Is X1 , X2 , . . . a Markov Chain?
Problem 27. Exercises 1, 2, 3, 5, 6, 13 in Chapter 4 of Ross.
Math 130B, Homework #1
Due: Wednesday, Jan 18 (in class)
Given some experiment, we let S denote the set of possible outcomes, and call subsets E S to be
events. Let denote the event consisting of no outcomes.
Basic manipulations
Recall the following axiom
Solutions for Homework 1.
Problem 1.
1. Note that the empty set is disjoint from itself, hence by the third probability axiom
P ( ) = P () + P ().
The left hand side is just P (), hence we have the relation that
P () = 2P ().
This is only satised if P ()
Math 130B, Homework #2
Due: January 27, in class
Bayes Theorem
Problem 1. Recall the Bayesian Spam Filter from class, where W1 is the event of seeing the word viagra
in a given message, and W2 is the event of seeing the wordenhancement (for example). Now