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a. Convolution B. using character ristic functions Let X, Y be i.i.d. r.v. with common distribution exp(lambda). Prove that W = X + Y has a gamma distribution with parameters (2, lambda) by using the following methods: Solution (a) \\( P(X+Y=z) = \\sum_{x=0}^z P(X=x) * P(Y=z-x) = \\sum_{x=0}^z \\lambda^2 e^{-\\lambda x} e^{-\\lambda(z-x)} \\) \\( P(X+Y=z) = z \\lambda^2 e^{-\\lambda z} \\) which is indeed gamma with \\( (2,\\lambda) \\) (b) \\( \\Phi_{X+Y}(t) = E(e^{itX})E(e^{itY})=(1-it/\\lambda)^{-2} \\) Which is indeeed the characteristic function of the gamma distribution with parameteters \\( (2,\\lambda) \\).

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- 1. a. Convolution B. using character ristic functions Let X, Y be i.i.d. r.v. with common distribution exp(lambda). Prove that W = X + Y has a gamma distribution with parameters (2, lambda) by using the following methods: Solution (a) ( P(X+Y=z) = sum_{x=0}^z P(X=x) * P(Y=z-x) = sum_{x=0}^z lambda^2 e^{-lambda x} e^{-lambda(z-x)} ) ( P(X+Y=z) = z lambda^2 e^{-lambda z} ) which is indeed gamma with ( (2,lambda) ) (b) ( Phi_{X+Y}(t) = E(e^{itX})E(e^{itY})=(1-it/lambda)^{-2} ) Which is indeeed the characteristic function of the gamma distribution with parameteters ( (2,lambda) )