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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: Ω = ℜ, or Ω = 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 Ω • 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 Ω 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. Chapter 4 Random Variables 64 • A random variable X is a function X : Ω →ℜ . Here, ℜ denotes the set of all real numbers • Every outcome in the sample space Ω is mapped onto a real number by the function X • The real number assigned to ϖ∈ Ω is denoted by X ( ϖ ) • Example: ϖ X ( ϖ ) 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) = 1–q • In this alternative formulation, the function X : Ω →ℜ is a fixed (i.e., nonrandom) map • ϖ∈ Ω always has the same real number X ( ϖ ) as its image • Randomness lies in which outcome ϖ ∈ Ω occurred, and not in the mapping – the mapping is unchanging...
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- Fall '06
- Probability theory, University of Illinois, Engineering Applications