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Homework Statement
Let X1, X2, … be independent exponentially distributed stochastic variables with parameter λ. For the sum Y = X1 + X2 + … + XN, where N is a geometrically distributed stochastic variable with parameter p, show that Y is exponentially distributed with parameter pλ.
Homework Equations
The expected value of exponentially distributed stochastic variables is 1/λ.
The expected value of geometrically distributed stochastic variables is 1/p.
The Attempt at a Solution
What I don’t understand is mainly why Y would be exponentially distributed. According to my notes, for N constant, Y would be Erlang(N,λ).
However, if we assume that the sum of exponentially distributed variables is exponentially distributed (how to prove this?), one should be able to calculate the expected value for this distribution as:
[tex]E(Y)=E(\sum_{i=1}^{N}X_{i})=E(\sum_{i=1}^{N} \underbrace{E(X_{i})}_{1/λ} ) =\frac{E(N)}{λ}=\frac{1}{pλ}[/tex]
Implying that Y~Exp(pλ). Is this correct?
So, I would appreciate any comments or hints hints that I could get from anyone here. I got the problem in Japanese and my Japanese isn't perfect so if the problem seems incorrect, feel free to point that out.
My first time posting here so I hope that I'm not breaking any rules.