View the step-by-step solution to:

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
Background image of page 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 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, 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
Background image of page 2
Sign up to view the entire interaction

Top Answer

Dear Student, I assessed your homework question, but unfortunately was not able to complete... View the full answer

Sign up to view the full answer

Why Join Course Hero?

Course Hero has all the homework and study help you need to succeed! We’ve got course-specific notes, study guides, and practice tests along with expert tutors.


Educational Resources
  • -

    Study Documents

    Find the best study resources around, tagged to your specific courses. Share your own to gain free Course Hero access.

    Browse Documents
  • -

    Question & Answers

    Get one-on-one homework help from our expert tutors—available online 24/7. Ask your own questions or browse existing Q&A threads. Satisfaction guaranteed!

    Ask a Question
Ask a homework question - tutors are online