Let X ∼ Exp(λ) and Y ∼ Exp(μ) be two independent exponential random variables, where λ, μ > 0. Find the probability density function of X + Y if λ ̸= μ.
I have successfully find ans if λ = μ, but stuck when finding pdf of X+Y if λ ̸= μ
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1 Answer
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The sum of exponential distributions each with it's own parameter turns out to be a hypoexponential distribution (https://en.wikipedia.org/wiki/Hypoexponential_distribution).
$$\sum_{i=1}^n \text{Exp}(\lambda_i) \sim \text{Hypoexponential}(\lambda_1,\dots,\lambda_n)$$
assuming $\lambda_i>0$.
answered May 23, 2022 at 8:46