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hw1prob2 - Homework Assignment 1 Queuing Theory 56 645558...

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Homework Assignment 1 Queuing Theory: 56 645558 01 Problem 2 - Sum of Random Variables Tony Smaldone [email protected] http://www.sntco.com/tony/masters.html July 26, 2001 Problem 2 - Sum of Random Variables Find the distribution (density) of Y = X 1 + X 2 for the two cases where a). X 1 Poisson ( λ 1 ) and X 2 Poisson ( λ 2 ) ; and, b). X 1 Exp ( λ 1 ) and X 2 Exp ( λ 2 ) . Assume independence. Solution a). Poisson Distribution Given two independent random variables X 1 Poisson ( λ 1 ) and X 2 Poisson ( λ 2 ) the question becomes what is the distribution of Y = X 1 + X 2 . Since X 1 and X 2 are independent events, the distribution of Y will be the sum of all possible values of X 1 and X 2 which sum to Y . For example, if P [ Y = k ], then P [ Y = k ] = P [ X 1 = 0] P [ X 2 = k ] + P [ X 1 = 1] P [ X 2 = k - 1] + · · · + P [ X 1 = k ] P [ X 2 = 0] = k n =0 P [ X 1 = n ] P [ X 2 = k - n ] Expanding the summation k n =0 P [ X 1 = n ] P [ X 2 = k - n ] 1
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= e - λ 1 λ 0 1 0! e - λ 2 λ k 2 k ! + e - λ 1 λ 1 1 1! e - λ 2 λ k - 1 2 k - 1!
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