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I have $ X_1, X_2 \stackrel{i.i.d.}{\sim} \mathrm{Exponential(\mu)}$, and their sum as a new random variabvle $ Y = X_1 + X_2 $. In order to calculate the PDF of Y,

\begin{align} \Pr[Y=y] &= \Pr[X_1=a,X_2=y-a] \\ &= \Pr[X_1 = a] \Pr[X_2 = y-a] \\ &= (\mu e^{-\mu a} )(\mu e^{-\mu (y-a) }) \\ &= \mu^2 e^{-\mu y} \end{align}

However, when I integrate $ \mu^2 e^{-\mu y} $ with respect to $y$, with $y>0$ the area under the curve exceeds 1. Is there an error in finding Y's PDF?

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Having calculated the probability for a particular $a$, you now need to integrate the result over all possible values of $a$ – between $0$ and $y$. This gives $\mu^2ye^{-\mu y}$, which does integrate to $1$ (it is a gamma distribution).

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    $\begingroup$ Makes sense! I was calculating $ \Pr[Y = y | X_1 = a] $, not $ \Pr[Y = y] $ $\endgroup$
    – ouiliame
    Aug 10, 2019 at 10:13

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