# Sum of exponential random variable with different means

Suppose that $X$ and $Y$ are independent exponential random variables with pdf's $f(x)=\lambda e^{-\lambda x}$ and $f(y) = \mu e^{- \mu y}$. What is $\;P \{ X+Y <t \}$ ie what is the cdf of the sum? I know that the distribution is gamma when the parameter is the same, but I'm not sure of a closed form when the parameters are different.

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Felix has provided a very nice detailed solution by hand. This type of problem can also be easily solved by automated methods using a computer algebra system ...

By independence, the joint pdf of $(X,Y)$ is say $f(x,y)$:

We seek the cdf of the sum, i.e. $P(X+Y<t)$:

where I am using the Prob function from the mathStatica add-on to Mathematica to calculate the probability automatically. All done.

Notes:

1. The answer is the same as that obtained by Felix.
2. As formal disclosure, I should perhaps add that I am one of the authors.
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Good to know this package exists. Thanks! –  mtiano Jan 12 at 17:31