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Math 130B, Homework #5 Due: February 17, in class Some review Problem 1. In class I showed that E(X ) = random variables n>0 P (X > n)...

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please do problem 3
2 pages 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, 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
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
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
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, 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

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This question was asked on Feb 14, 2012 and answered on Feb 14, 2012.

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