convol_ - Sums of Independent Random Variables Consider the...

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Sums of Independent Random Variables Consider the sum of two independent discrete random variables X and Y whose values are restricted to the non-negative integers. Let f X ( · ) denote the probability distribution of X and f Y ( · ) denote the probability distribution of Y . The distribution of their sum Z = X + Y is given by the discrete convolution formula . Theorem Discrete Convolution Formula. The random variable Z = X + Y has probability distribution f Z ( · ) given by f Z ( z ) = f X + Y ( z ) = P ( Z = z ) = z X x =0 f X ( x ) f Y ( z - x ) for z = 0 , 1 , ... . Proof: For each z , the event [ Z = z ] is the union of the disjoint events [ X = x and Y = z - x ] for x = 0 , 1 , ..., z . Consequently, P ( Z = z ) = f Z ( z ) = z x =0 P ( X = x and Y = z - x ) = z x =0 f X ( x ) f Y ( z - x ) where the last step follows by independence. Let X 1 and X 2 be independent binomial random variables having the same probability of success. Their sum is again binomial. Corollary 1
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This note was uploaded on 10/20/2011 for the course AMS 310 taught by Professor Mendell during the Spring '08 term at SUNY Stony Brook.

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convol_ - Sums of Independent Random Variables Consider the...

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