Chapter 4: One Random Variable

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering

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Chapter 4: One Random Variable 4.1

The Cumulative Distribution Function

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© 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.


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The Probability Density Function

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The Expected Value of X

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Important Continuous Random Variables

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A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 4.62

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a)

Using Tables 4.2 and 4.3:



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(a)

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Functions of a Random Variable

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The Markov and Chebyshev Inequalities

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Transform Methods

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Basic Reliability Calculations

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Computer Methods for Generating Random Variables The following Octave commands generate the requested plots:

(a) x = [-4:0.01:4]; y0 = normal_pdf(x, -2, 1); y1 = normal_pdf(x, 2, 1); figure; hold on; plot(x, y0, "1"); plot(x, y1, "3");

(b) x = [-5:0.01:5]; y0 = 1-normal_cdf(x, -2, 1); y1 = 1-normal_cdf(x, 2, 1); ey0 = e.^(-(x+2).^2/2); ey1 = e.^(-(x-2).^2/2); figure; hold on; plot(x, y0, "1"); plot(x, y1, "3"); plot(x, ey0, "1"); plot(x, ey1, "3");



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The following Octave commands generate plots of the pdf and cdf of the gamma random variable:

(a) x = [0:0.01:15]; figure; hold on; plot(x, gamma_pdf(x, plot(x, gamma_pdf(x, plot(x, gamma_pdf(x, figure; hold on; plot(x, gamma_cdf(x, plot(x, gamma_cdf(x, plot(x, gamma_cdf(x,

1, 1), "1"); 2, 1), "2"); 4, 1), "3");

1, 1), "1"); 2, 1), "2"); 4, 1), "3");

(b) x = [0:0.01:15]; figure; hold on; plot(x, gamma_pdf(x, plot(x, gamma_pdf(x, plot(x, gamma_pdf(x, plot(x, gamma_pdf(x, figure; hold on; plot(x, gamma_cdf(x, plot(x, gamma_cdf(x, plot(x, gamma_cdf(x, plot(x, gamma_cdf(x,

1/2, 1/2), "1"); 1, 1/2), "2"); 3/2, 1/2), "3"); 5/2, 1/2), "4");

1/2, 1/2), "1"); 1, 1/2), "2"); 3/2, 1/2), "3"); 5/2, 1/2), "4");



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The following Octave commands generate the requested samples and plots:

(a) x = [-6:0.01:6]; u = rand(1, 1000); %Multiply all values by discretely generated -1 or 1 z = -log(u).*discrete_rnd(length(u), [-1 1], [0.5 0.5]); figure; hold on; %Normalize to 2 because bar width is 0.5 hist(z, [-6:0.5:6], 2); plot(x, laplace_pdf(x), "1");

(b) x = [1:0.01:10]; u = rand(1, 1000); k = 1.5; z = u.^(-1/k); figure; hold on; hist(z, [1.25:0.5:10], 2); plot(x, k./x.^(k+1)); x = [1:0.01:10]; u = rand(1, 1000); k = 2; z = u.^(-1/k); figure; hold on; hist(z, [1.25:0.5:10], 2); plot(x, k./x.^(k+1)); x = [1:0.01:10]; u = rand(1, 1000); k = 2.5; z = u.^(-1/k); figure; hold on; hist(z, [1.25:0.5:10], 2); plot(x, k./x.^(k+1));

(c) x = [0:0.01:5]; u = rand(1, 1000); b = 0.5; z = log(1./u).^(1./b); figure; hold on; hist(z, [0:0.25:5], 4); plot(x, b.*x.^(b-1).*e.^(-x.^b), "1"); x = [0:0.01:5]; u = rand(1, 1000); © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.


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b = 2; z = log(1./u).^(1./b); figure; hold on; hist(z, [0:0.125:5], 8); plot(x, b.*x.^(b-1).*e.^(-x.^b), "1"); x = [0:0.01:5]; u = rand(1, 1000); b = 3; z = log(1./u).^(1./b); figure; hold on; hist(z, [0:0.125:5], 8); plot(x, b.*x.^(b-1).*e.^(-x.^b), "1");



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The following Octave commands describe the function for performing the rejection method and the code to call the function: function z = gaussian_rejection_method(N) z = zeros(1, N); k = 1; while k