Chapter 4 - Chapter 4 Random Variables Random variables are...

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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: = ℜ, 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 0 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 co-domain A function f with domain A and co-domain 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 pre-image of y. We write f(x) = y Every x A has an image f(x) B Every y B need not have a pre-image in A An element y B might have more than one pre-image in A The set of all y B that have pre-images in A is called the range of f In the picture shown below, the range is the shaded oval-shaped set. Domain Co-domain © 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 4 Random Variables 64 A random variable X is a function X : →ℜ . 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 ϖ∈ is denoted by X ( ϖ ) Example: ϖ X ( ϖ ) HH 2 HT 1 TH 1 TT 0 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) = 1–q In this alternative formulation, the function X : →ℜ is a fixed (i.e., nonrandom) map ϖ∈ always has the same real number X ( ϖ ) as its image Randomness lies in which outcome ϖ occurred, and not in the mapping – the mapping is unchanging The observed value X ( ϖ ) depends on which ϖ
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