chapter 5 - Chapter 5 Many Random Variables A random...

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Chapter 5 Many Random Variables A random variable models phenomena in which the experimental outcomes are numbers A random variable measures one physical parameter Different random variables measure different parameters Example: Requests for different files arrive at a Web server. Requests have two parameters of interest: the time of arrival and the length of the data file requested Sunspot activity causes different levels of fading of radio signals in different frequency bands Different random variables measure different parameters Experimental observations yield vectors or sequences or arrays of numbers The observed vectors are random: we cannot predict beforehand which vector will be observed next The mathematical model of a random variable associates numbers with outcomes ϖ in a sample space Different random variables associate different numbers with each outcome ϖ [ X ( ϖ ), Y ( ϖ ), Z ( ϖ ), … ] is a vector of observations Example: = set of people. One is chosen at random X = height of person chosen Y = weight of person chosen ( X , Y ) is a random vector or a bivariate random variable ( X , Y ) maps ϖ onto the pair of numbers ( X ( ϖ ), Y ( ϖ )) ( X , Y ) maps ϖ onto point ( X ( ϖ ), Y ( ϖ )) in the plane Coordinate axes are u and v with value of X plotted along u axis and of Y along v axis u v X takes on three values, Y takes on only two values Both X and Y individually are random variables Together , X and Y create a joint mapping of ϖ onto the random point ( X ( ϖ ), Y ( ϖ )) As usual, we drop the explicit dependence on ϖ when talking about random variables ( X , Y ) is a random point in the plane On succesive trials of the experiment, the outcomes are ϖ 1 , ϖ 2 , ϖ 3, and we observe the random points ( X ( ϖ 1 ), Y ( ϖ 1 )), ( X ( ϖ 2 ), Y ( ϖ 2 )), ( X ( ϖ 3 ), Y ( ϖ 3 )), … The u-coordinate of the random point is X ( ϖ ) and the v-coordinate is Y ( ϖ ) © 1997 by Dilip V. Sarwate. All rights reserved. No part of this manuscript may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without the prior written permission of the author.
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Chapter 5 Many Random Variables 115 The random point ( X , Y ) has more information than either X or Y ; it describes the joint behavior of these random variables Example: Let X and Y be discrete random variables with P{ X = 0} = { Y = 0} = 1/2 and P{ X = 1} = { Y = 1} = 1/2 What is the probability that the random point ( X , Y ) has value (1,1)? that is, what is P({ X = 1} { Y = 1})? Given events A and B with probabilities P(A) and P(B), what is P(A B)? We might have P({ X = 1} { Y = 1}) = 1/4; P({ X = 1} { Y = 0}) = 1/4 P({ X = 0} { Y = 1}) = 1/4; P({ X = 0} { Y = 0}) = 1/4 We might have P({ X = 1} { Y = 1}) = 1/3; P({ X = 1} { Y = 0}) = 1/6; P({ X = 0} { Y = 1}) = 1/6; P({ X = 0} { Y = 0}) = 1/3 Thus, the individual behavior of X and Y cannot completely specify their joint behavior The joint CDF of X and Y , or the CDF of the bivariate random variable ( X , Y ), or the joint CDF of the random vector ( X , Y
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