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please do problem 3
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please do problem 3
Homework5.pdf

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 this expression to show


P (X n).

E(X ) =
n=1

2. Using a similar idea, show that




P (X j, Y k ).

E(XY ) =
j =1 k=1

Problem 2. Let X Poisson(). Show that P (X = j ) monotonically increases as j increases for 0 j ,
and then monotonically decreases for j > . Note that may not be an integer. (Hint consider the ratios
P (X = j )/P (X = j 1).)
Problem 3. Show that a Poisson() random variable X is concentrated around its mean in the following
sense: For any

>0
P(

|X |
> ) 0, as .


Problem 4. A large number N of people are subjected to a blood test. This can be administered in two
ways:
1. Each person can be tested separately, in this case N test are required,
2. The blood samples of k persons can be pooled and analyzed together. If this test is negative, this one
test suces for the k people. If the test is positive, each of the k persons must be tested separately,
and in all, k + 1 tests are required for the k people.
Assume that the probability p that a test is positive is the same for all people and that these events are
independent. For small p, show that the value of k which will minimize the expected number of tests under
1
the second plan is approximately p .
1

More conditional probability
Problem 5. Suppose we continually roll a die until the sum of all throws exceeds 100. What is the most
likely value of this total when you stop? (Hint: condition on the total achieved before the last throw).
Problem 6. Suppose you go to Vegas with D dollars in your pocket, and repeatedly bet a dollar at a game
with win probability p. If you win you gain a dollar, if you lose you lose a dollar. You quit the game when
you run out of dollars.
1. Suppose D = 1. Use the Ballot Theorem to show that the probability of quitting at round 2k + 1 is
2k + 1
1
pk (1 p)k+1 .
k + 1 2k + 1
2. Now suppose D is a positive even integer. Compute the probability of quitting at round 2k .
3. (Optional, but highly recommended). Go to www.wolframalpha.com, and input the following line,
trying various values of p between 0 and 1. (including 1/2).
sum (Binom(2k+1, k+1))/(2k+1) * p^k*(1-p)^(k+1) from 0 to infinity
What does this tell you about probability of eventually quitting?

Problem 7. Complete the following alternative proof of the Ballot Theorem. Recall that Pn,m is the
probability of candidate A always leading candidate B, given n votes for A and m votes for B.
1. Argue that
Pn,m = 1 P (A and B are tied at some point).
2. Explain why
P (A receives rst vote and they are eventually tied)
= P (B receives rst vote and they are eventually tied).
(Hint: look at the representations of votes as paths, like in class, and show a 1-1 correspondence. This
part is known as the reection principle, for a good reason!)
3. Show that P (eventually tied) = 2m/(n + m), and then by part 1 conclude that
Pn,m =

2

nm
.
n+m

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