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) \\).
a. ConvolutionB. using character ristic functions Let X, Y be i.i..pdf
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) )
