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Unformatted text preview: Chapter 4 Random Variables Random variables are used to model phenomena in which the experimental outcomes are numbers , e.g. 1, 2, 3, or 3.213678 instead of labels such as Head or Tail or Luke or Darth Example: = , or = Z We do not know for sure which number will be observed when the experiment is performed; only that it is some number in the sample space X denotes the random number that we observe. It is called a random variable. A different number (value of X ) is typically observed on each trial of the experiment Hence variable Number X is an outcome of a random experiment Hence random Alternative formulation A random variable X is obtained by associating real numbers with the outcomes of random experiments We have already used this idea before, but we did not bother to give it a formal name Example: The experiment consists of tossing a coin twice. X denotes the number of heads observed Outcome Value of X HH 2 HT 1 TH 1 TT In this alternative formulation, X is also thought of as the function or mapping that maps into real numbers Digression for brief review A function is a mapping from a set called the domain to a set called the codomain A function f with domain A and codomain B is denoted f: A B Every x A is associated with some y B y is the image of x, and x is the preimage of y. We write f(x) = y Every x A has an image f(x) B Every y B need not have a preimage in A An element y B might have more than one preimage in A The set of all y B that have preimages in A is called the range of f In the picture shown below, the range is the shaded ovalshaped set. Domain Codomain 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. Chapter 4 Random Variables 64 A random variable X is a function X : . Here, denotes the set of all real numbers Every outcome in the sample space is mapped onto a real number by the function X The real number assigned to is denoted by X ( ) Example: X ( ) HH 2 HT 1 TH 1 TT There is nothing random about this function. It always maps HH onto 2, for example The randomness arises from the fact that we do not know which of the 4 outcomes will occur, and hence we do not know which of the 3 numbers 0, 1, 2 will be observed The numbers 0, 1, 2 occur with probabilities q 2 , 2pq and p 2 respectively; p = P(H) = 1q In this alternative formulation, the function X : is a fixed (i.e., nonrandom) map always has the same real number X ( ) as its image Randomness lies in which outcome occurred, and not in the mapping the mapping is unchanging...
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This test prep was uploaded on 09/28/2007 for the course BTRY 4080 taught by Professor Schwager during the Fall '06 term at Cornell University (Engineering School).
 Fall '06
 SCHWAGER

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