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, let’s deﬁne 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(λ). Show that P (X = j ) monotonically increases as j increases for 0 ≤ j ≤ λ,

and then monotonically decreases for j > λ. Note that λ may not be an integer. (Hint consider the ratios

P (X = j )/P (X = j − 1).)

Problem 3. Show that a Poisson(λ) random variable X is concentrated around its mean λ in the following

sense: For any >0

P( |X − λ|

> ) → 0, as λ → ∞.

λ 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 suﬃces 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 ﬁrst vote and they are eventually tied)

= P (B receives ﬁrst 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 reﬂection principle, for a good reason!)

3. Show that P (eventually tied) = 2m/(n + m), and then by part 1 conclude that

Pn,m = 2 n−m

.

n+m