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# lecturenotesweek8 - ISyE 2027 Probability with Applications...

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ISyE 2027 Probability with Applications Polly B. He Fall10, Week 8 Reading: Section 7.3-7.4 The Change-of-Variable Formula If we know the distribution of X , we can ﬁnd its expected value E( X ) if it exists. What if I want to ﬁnd the expected value of g ( X ) where g ( · ) is a real-valued function, e.g., g ( X ) = X 2 ? Example 7.3.1 Suppose U U (0 , 1). We want to ﬁnd E ( U 2 ). The brute-force approach: 1

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The “Smart” approach: The change-of-variable formula: Let X be a random variable, and let g : R R be a function. If X is continuous with probability density function f , then E[ g ( X )] = -∞ g ( x ) f ( x ) dx If X is discrete, taking the values a 1 , a 2 , . . . , then E[ g ( X )] = i g ( a i ) P ( X = a i ) Continue with Example 7.3.1, Note: 1. In general, E( X 2 ) ̸ = (E( X )) 2 . 2. If you set g ( x ) = x in the change-of-variable formula, you will recover the mean. Variance
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## This note was uploaded on 02/21/2011 for the course ISYE 2027 taught by Professor Zahrn during the Fall '08 term at Georgia Tech.

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lecturenotesweek8 - ISyE 2027 Probability with Applications...

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