# The smallest possible value of s is 2 and s 1 2 1

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Unformatted text preview: t either occurs or does not occur. A natural and useful generalization allows for more than two values: Deﬁnition 2.1.1 A random variable is a real-valued function on Ω. Thus, if X is a random variable for a probability space (Ω, F , P ), if the probability experiment is performed, which means a value ω is selected from Ω, then the value of the random variable is X (ω ). Therefore, a random variable can take many diﬀerent values, and for each possible value there is some probability that the random variable is equal to that value. A random variable is said to be discrete-type if there is a ﬁnite set u1 , . . . , un or a countably inﬁnite set u1 , u2 , . . . such that P {X ∈ {u1 , u2 , . . .}} = 1. (2.1) The probability mass function (pmf) for a discrete-type random variable X, pX , is deﬁned by pX (u) = P {X = u}. Note that (2.1) can be written as: pX (ui ) = 1. i The pmf is suﬃcient to determine the probability of any event determined by X , because for any set A, P {X ∈ A} = i:ui ∈A pX (ui ). Example 2.1.2 Let...
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## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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