# week10 - Example A device containing two key components...

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week 10 1 Example A device containing two key components fails when and only when both components fail. The lifetime, T 1 and T 2 , of these components are independent with a common density function given by The cost, X , of operating the device until failure is 2 T 1 + T 2 . Find the density function of X . () > = otherwise t e t f t T 0 0

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week 10 2 Convolution Suppose X, Y jointly distributed random variables. We want to find the probability / density function of Z=X+Y . Discrete case X, Y have joint probability function p X,Y ( x,y ). Z = z whenever X = x and Y = z – x . So the probability that Z = z is the sum over all x of these joint probabilities. That is If X, Y independent then This is known as the convolution of p X ( x ) and p Y ( y ). ( ) ( ) = x Y X Z x z x p z p . , , ( ) ( ) ( ) = x Y X Z x z p x p z p .
week 10 3 Example Suppose X ~ Poisson( λ 1 ) independent of Y ~ Poisson( λ 2 ). Find the distribution of X+Y .

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week 10 4 Convolution - Continuous case Suppose X, Y random variables with joint density function f X,Y ( x,y ). We want to find the density function of Z=X+Y . Can find distribution function of Z and differentiate. How? The Cdf of Z can be found as follows: If is continuous at z then the density function of Z is given by If X, Y independent then This is known as the convolution of f X ( x ) and f Y ( y ). () ( ) ( ) () ∫∫ −∞ = −∞ = −∞ =− = −∞ = −∞ = = = = + = z vx Y X x z v Y X x x z y Y X Z dxdv x v x f dvdx x v x f dydx y x f z Y X P z F . , , , , , , −∞ = x XY dx x v x f , () ( ) −∞ = = x XY Z dx x z x f z f , () () ( ) −∞ = = x Y X Z dx x z f x f z f
week 10 5 Example X, Y independent each having Exponential distribution with mean 1/ λ . Find the density for W=X+Y.

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week 10 6 Some Recalls on Normal Distribution If Z ~ N (0,1) the density of Z is If X = σ Z + μ then X ~ N ( μ , σ 2 ) and the density of X is If X ~ N ( μ , σ 2 ) then () < < =
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## This note was uploaded on 09/22/2010 for the course STATISTICS STA257 taught by Professor Mcdunnough during the Fall '10 term at University of Toronto- Toronto.

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week10 - Example A device containing two key components...

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