Chapter 5: Pairs of Random Variables

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

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Chapter 5: Pairs of Random Variables 5.1

Two Random Variables

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Pairs of Discrete Random Variables

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The Joint pdf of X and Y

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The Joint cdf of Two Continuous Random Variables

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5.36

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5.36b

5.36c

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Independence of Two Random Variables

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5.6 Joint Moments and Expected Value of a Function of Two Random Variables 5.56

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Conditional Probability and Conditional Expectation

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9

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

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Functions of Two Random Variables

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

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(b)

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5.9

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Pairs of Jointly Gaussian Random Variables

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5.114 (a)

(b)

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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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© 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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5.10 Generating Independent Gaussian Random Variables 5.121

The following Octave code produces the inverse:

function z = rayleigh_rnd(s) u = rand; z = s.*(2.*log(1./(1-u))).^(1/2); end

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The following Octave generates the requested pairs and plot:

len = 10000; X = discrete_rnd([1 -1], [0.5 0.5], 2, len); N = normal_rnd(0, 1, 2, len); Y = X + N; figure; plot(Y(1,:), Y(2,:), "."); Xr = sign(Y); sigerr = (X ~= Xr); biterr = (sigerr(1,:) | sigerr(2,:)); proberr = sum(biterr)./len

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The following Octave generates the requested pairs and plot:

X = normal_rnd(0, 2, 1, 1000); N = normal_rnd(0, 1, 1, 1000); Y = X + N; Xr = Y./(1 + 1/2); err = Xr - X; figure; hist(err, [-3:0.25:3], 4); m = mean(err) v = var(err)

© 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.

A. Leon-Garcia INSTRUCTOR’S SOLUTIONS MANUAL Probability, Statistics, and Random Processes for Electrical Engineering 5.125

5-120

The following Octave generates the sequence of Xn and Yn:

X = normal_rnd(0, 1, 1, 1000); Y = (X + [0 X(1:999)])./2; figure; plot(Y(1:999), Y(2:1000), "."); Z = (X - [0 X(1:999)])./2; figure; plot(Z(1:999), Z(2:1000), ".");

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The following Octave generates the specified jointly Gaussian random

variables: function vw = gaussian_correlate(xy, mu1, mu2, var1, var2, rho) sig1 = sqrt(var1); sig2 = sqrt(var2); K = [var1, rho*sig1*sig2; rho*sig1*sig2, var2]; [evec eval] = eig(K); A = evec * sqrt(eval); vw = zeros(size(xy)); for i = 1:length(xy) vw(:,i) = A*xy(:,i) + [mu1; mu2]; end end

© 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 following Octave generates the specified jointly Gaussian random 5.127 variables and plot: xy = normal_rnd(0, 1, 2, 1000); vw = gaussian_correlate(xy, 1, -1, 1, 2, -1/2); plot(vw(1,:), vw(2,:), ".");

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The following Octave generates the specified plot:

xy = normal_rnd(0, 1, 2, 1000); muh = 174; muv = 4.4; varh = 42.36; varv = 0.021; covhv = 0.458; sigh = sqrt(varh); sigv = sqrt(varv); rhohv = covhv/(sigh*sigv); xy = normal_rnd(0, 1, 2, 1000); hv = zeros(size(xy)); hv = gaussian_correlate(xy, muh, muv, varh, varv, rhohv); hw = zeros(size(hv)); hw(1,:) = hv(1,:); hw(2,:) = e.^hv(2,:); bmi = hw(2,:)./(hw(1,:).^2); hist(bmi);

© 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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Problems Requiring Cumulative Knowledge 5.129

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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.