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$X_1$ and $X_2$ are i.i.d random variables and the pdf of each of them is $e^{-x}$ for $x>0$ and $0$ otherwise. $Y = X_1-X_2$ and the question asks to find the pdf for $Y$? I took the approach of going from the cdf to pdf. $P(Y\le y) = P(X1-X2\le y) = P(X1\le y+X2)$. The integral boundaries would be different, depending on the sign of $y$. The integrand is $e^{-X_1-X2}$ and I did it in the order $dX_1dX_2$.

In the case of $y>$0 I set the boundary for $dX_1$ from $0$ to $y+X_2$ and $0$ to $\infty$ for $dX_2$. For $y\le0$ I have $dX_1$ set from $0$ to $y+X_2$. And the boundary for $dX2$ from $-y$ to $\infty$. I tend to make mistakes when setting boundaries for these types of piecewise functions, so was wondering if I made any here? Before I proceed any further in this problem.

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