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4.139 The moment-generating function of a normally distributed random variable, μ and variance o² was shown in Exercise 4.138 to be m(t) et+ (1/2)²2. in Exercise 4.137 to derive the moment-generating distribution of X? Why? Y, with mean Use the result = function of X = -3Y + 4. What is the

4.139 The moment-generating function of a normally distributed random variable, μ and variance o² was shown in Exercise 4.138 to be m(t) et+ (1/2)²2. in Exercise 4.137 to derive the moment-generating distribution of X? Why? Y, with mean Use the result = function of X = -3Y + 4. What is the

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Pls how do i solve 4.139?
4.136 4.139 4.140 I can be used as numerical descriptive measures to describe the data that we obtain in The purpose of the preceding discussion of moments is twofold. First, moments variable possesses a particular probability distribution. It can be shown that if two an experiment. Second, they can be used in a theoretical sense to prove that a random random variables Y and Z possess identical moment-generating functions, then Y and Z possess identical probability distributions. This latter application of moments was mentioned in the discussion of moment-generating functions for discrete random variables in Section 3.9; it applies to continuous random variables as well. For your convenience, the probability and density functions, means, variances, and moment-generating functions for some common random variables are given in Appendix 2 and inside the back cover of this text. Exercises Suppose that the waiting time for the first customer to enter a retail shop after 9:00 A.M. is a random variable Y with an exponential density function given by y > 0, a {6). f(y) = e-x/e 4.138 Example 4.16 derives the moment-generating function for Yu, where Y is normally dis- tributed with mean μ and variance o². elsewhere. a Find the moment-generating function for Y. b Use the answer from part (a) to find E(Y) and V (Y). Show that the result given in Exercise 3.158 also holds for continuous random variables. That is, show that, if Y is a random variable with moment-generating function m(t) and U is given by U = aY + b, the moment-generating function of U is ebm (at). If Y has mean and variance o², use the moment-generating function of U to derive the mean and variance of U. Use the results in Example 4.16 and Exercise 4.137 to find the moment-generating function for Y. a m(t) = (1-4t)-². b m(t)=1/(1-3.2t). c m(t) = e-51+61² b Differentiate the moment-generating function found in part (a) to show that E(Y) = μ and V(Y) = 0². The moment-generating function of a normally distributed random variable, Y, with mean μ and variance o2 was shown in Exercise 4.138 to be m(t) = et+ (1/2)²o². Use the result in Exercise 4.137 to derive the moment-generating function of X = -3Y + 4. What is the distribution of X? Why? Identify the distributions of the random variables with the following moment-generating functions:
Transcribed Image Text:4.136 4.139 4.140 I can be used as numerical descriptive measures to describe the data that we obtain in The purpose of the preceding discussion of moments is twofold. First, moments variable possesses a particular probability distribution. It can be shown that if two an experiment. Second, they can be used in a theoretical sense to prove that a random random variables Y and Z possess identical moment-generating functions, then Y and Z possess identical probability distributions. This latter application of moments was mentioned in the discussion of moment-generating functions for discrete random variables in Section 3.9; it applies to continuous random variables as well. For your convenience, the probability and density functions, means, variances, and moment-generating functions for some common random variables are given in Appendix 2 and inside the back cover of this text. Exercises Suppose that the waiting time for the first customer to enter a retail shop after 9:00 A.M. is a random variable Y with an exponential density function given by y > 0, a {6). f(y) = e-x/e 4.138 Example 4.16 derives the moment-generating function for Yu, where Y is normally dis- tributed with mean μ and variance o². elsewhere. a Find the moment-generating function for Y. b Use the answer from part (a) to find E(Y) and V (Y). Show that the result given in Exercise 3.158 also holds for continuous random variables. That is, show that, if Y is a random variable with moment-generating function m(t) and U is given by U = aY + b, the moment-generating function of U is ebm (at). If Y has mean and variance o², use the moment-generating function of U to derive the mean and variance of U. Use the results in Example 4.16 and Exercise 4.137 to find the moment-generating function for Y. a m(t) = (1-4t)-². b m(t)=1/(1-3.2t). c m(t) = e-51+61² b Differentiate the moment-generating function found in part (a) to show that E(Y) = μ and V(Y) = 0². The moment-generating function of a normally distributed random variable, Y, with mean μ and variance o2 was shown in Exercise 4.138 to be m(t) = et+ (1/2)²o². Use the result in Exercise 4.137 to derive the moment-generating function of X = -3Y + 4. What is the distribution of X? Why? Identify the distributions of the random variables with the following moment-generating functions:
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