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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)...

please do problem 3
Math 130B, Homework #5 Due: February 17, in class Some review Problem 1. 1. In class I showed that E ( X ) = n> 0 P ( X > n ) by just writing out the sum. Instead, let’s define the random variables I n = ± 1 if X n 0 if X < n and express X in terms of the I n ’s. Use this expression to show E ( X ) = X n =1 P ( X n ) . 2. Using a similar idea, show that E ( XY ) = X j =1 X k =1 P ( X j,Y k ) . 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 | λ > ± ) 0 , as 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 the second plan is approximately 1 p . 1
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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 2 k + 1 is ± 2 k + 1 k + 1 ² 1 2 k + 1 p k (1 - p ) k +1 . 2. Now suppose D is a positive even integer. Compute the probability of quitting at round 2 k . 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 P n,m is the probability of candidate A always leading candidate B, given n votes for A and m votes for B. 1. Argue that P n,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) = 2 m/ ( n + m ), and then by part 1 conclude that P n,m = n - m n + m . 2
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