probability

User Generated

jnatqnyh

Mathematics

Description

solve the problem in specific process

the questions contain several kinds of distribution like joint, gamma, normal


Unformatted Attachment Preview

SONY MSI POD 2GB APPLIED PROBLEMS 1. An urn contains r = n1 + r2 packages of candy; of these, i contain one piece and r2 contain three pieces. (a) If a package were to be drawn from the urn at random, what is the probability it would contain one piece of candy? Three pieces? If two packages were drawn (without replacement), what are the probabilities of obtaining altogether two, four, or six pieces of candy? For (b) and (c) below, we suppose that a sample of n packages (n 0 3e-3y, if y 20, fx (x) = fy(y) = 0, otherwise; 0, otherwise. urn Find the density function Sz(2) for Z = X + Y (a) by computing Fz(z); (b) using (3.2) of Chapter 6. 7. Let X1, X2,..., X 25 be independent random variables, each of which is uniformly distributed on the interval (0,2), and let X = Lisi Xi. (a) Find the mean and variance of X. (b) Use the central limit theorem to estimate the probability that (X - 24 < 3. (c) Use Chebyshev's inequality to find a number a such that you are absolutely sure that P{ X - E[X]| a} 5) exactly. Find its upper bound by using Chebyshev's inequality. 12. "I've always believed that 10% of the financiers on Wall Street are shady," grumbled the S.E.C. inspector, "but the difficulty is to know which ones they are. Take the Amalgamated Widgets crash. An honest investor would have had only a 20% chance of coming out unscathed, but there is a 70% chance that a crook would have had inside information and gotten out in time. Now Smith didn't lose any money ...." Assuming that the inspector's estimates are correct, what is the probability that Smith is honest? 13. A large urn contains N balls of each of 20 different colors (that is, a total of 20N balls). 10 balls are selected at random; we let X be the total number of different colors obtained, and write X = Liel, with X, a Bernoulli random variable indicating whether or not the ith color is obtained. (a) Suppose that the selection is without replacement. Find E[X] and Var(X). (Hint: compute E[X] by computing P{Xi = 0} using equally likely outcomes; do E[X;X;] similarly.) (b) Find by elementary reasoning the correct values in (a) when N = 1, and verify that the answers you obtained there are correct in that case. (c) Suppose that the selection is with replacement. Find E(X) and Var(X). Your answer should not depend on N. (d) Show that your answers in (c) are the N o limit of your answers in (a). Explain why this should be true. 14. Each morning, Fred makes a random choice of one of three routes to take to work. After n trips, (n > 0), what is the probability that he has traveled each route at least once? Hint: use inclusion-exclusion to calculate the probability of the complementary event. 15. Explain the coupon collecting problem. 16. ## 6.47 - 6.49 17. #8.11 Please review homework problems, especially the last two sets!
User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

This question has not been answered.

Create a free account to get help with this and any other question!

Related Tags