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Lecture_10

# Lecture_10 - 1 EXPECTATION Lecture 10 ORIE3500/5500...

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1 EXPECTATION Lecture 10 ORIE3500/5500 Summer2009 Chen Class Today Expectation 1 Expectation 1.1 Expectation of a Random Variable Expectation or expected value of a random variable denoted by E ( X ) of just EX is the weighted average of the values it takes, where the weight are the chances of the random variable taking that value. As the name suggests, if we were to expect some value from a random variable to take, then this would be the one. The expected value of a discrete random variable X , with pmf p X , is defined as E ( X ) = X i x i p X ( x i ) . For a continuous random variable X with density f X , the expectation is defined as E ( X ) = Z -∞ xf X ( x ) dx. Sometimes it is difficult to compute the expectation of a random vari- able by following the definition. There are alternate approaches to compute expectation. The next example deals with one such case. Example If X is discrete and takes values 0 , 1 , 2 , . . . , then we can reduce the expression of expectation involving the cdf to E ( X ) = X n =0 P [ X > n ] = X n =0 [1 - F X ( n )] . It is not difficult to see why this is true. As in the continuous case this 1

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1.1 Expectation of a Random Variable 1 EXPECTATION involves changing of the order of summation. E ( X ) = X n =0 np X ( n ) = X n =0 nP [ X = n ] = P [ X = 1] + P [ X = 2] + P [ X = 2] + P [ X = 3] + P [ X = 3] + P [ X = 3] . . . = P [ X = 1] + P [ X = 2] + P [ X = 3] + P [ X = 4] + · · · + P [ X = 2] + P [ X = 3] + P [ X = 4] + · · · + P [ X = 3] + P [ X = 4] + · · · .
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Lecture_10 - 1 EXPECTATION Lecture 10 ORIE3500/5500...

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