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Unformatted text preview: m, X + Y. This section focuses on determining the distribution of the sum, X + Y, under various assumptions on the joint distribution of X and Y. 4.5.1 Sums of integer-valued random variables Suppose S = X + Y, where X and Y are integer-valued random variables. We shall derive the pmf of S in terms of the joint pmf of X and Y. For a fixed value k , the possible ways to get S = k can be indexed according to the value of X. That is, for S = k to happen, it must be that X = j and Y = k − j for some value of j. Therefore, by the law of total probability, pS (k ) = P {X + Y = k } P {X = j, Y = k − j } = j pX,Y (j, k − j ). = (4.13) j If X and Y are independent, then pX,Y (j, k − j ) = pX (j )pY (k − j ), and (4.13) becomes: pX (j )pY (k − j ). pS (k ) = (4.14) j By the definition of the convolution operation “∗”, (4.14) is equivalent to: pS = pX ∗ pY (if S = X + Y , where X , Y are independent). 1 If the density did factor, then for 0 < u1 < u2 < 1 and 0 < v1 < v2 &l...
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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 Tech.

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