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lect12

# lect12 - EXPECTATION OF SUMS OF RANDOM VARIABLES SOME...

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EXPECTATION OF SUMS OF RANDOM VARIABLES Outline EXPECTATION OF SUMS OF RANDOM VARIABLES COUPON COLLECTION SOME USEFUL FACTS FROM ONE-VARIABLE CALCULUS 1 / 15 Xinghua Zheng Lect 12: Expectation of sums; and some calculus

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EXPECTATION OF SUMS OF RANDOM VARIABLES COUPON COLLECTION THE ADDITION RULE FOR EXPECTATION Have seen and used frequently the Addition Rule for expectation: If X and Y are random variables then E ( X + Y ) = E ( X ) + E ( Y ) 1 This holds no matter whether X and Y are independent or not! In general, for any random variables X 1 , . . . , X n , E ( X 1 + . . . + X n ) = E ( X 1 ) + . . . + E ( X n ) . 1 see pp. 165 - 166 in Ross if you’d like to know the proof. 2 / 15 Xinghua Zheng Lect 12: Expectation of sums; and some calculus
EXPECTATION OF SUMS OF RANDOM VARIABLES COUPON COLLECTION EXAMPLES If X B ( n , p ) , then X = X 1 + . . . + X n , where X i = , and E ( X ) = If X has a negative binomial ( r , p ) distribution, then X = X 1 + . . . + X r , where X i = , and E ( X ) = An urn contains G good items and B bad items, for a total of N = G + B items. n items will be drawn at random without replacement from the urn. Let X be the number of good items in the sample. Have seen that X has a hypergeometric distribution. Moreover, X = X 1 + . . . + X n , where X i = , and E ( X ) = with p := G / N . In this example, X i ’s are (independent/dependent?) of each other. 3 / 15 Xinghua Zheng Lect 12: Expectation of sums; and some calculus

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EXPECTATION OF SUMS OF RANDOM VARIABLES COUPON COLLECTION EXAMPLES, ctd An urn contains G good items and B bad items, for a total of N = G + B items. n items will be drawn at random without replacement from the urn. Let X be the number of good items in the sample. What’s E ( X 2 ) ?
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lect12 - EXPECTATION OF SUMS OF RANDOM VARIABLES SOME...

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