# Find PDF of $Y = X_1-X_2$ [duplicate]

$$X_1$$ 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.

• You have done it correctly. Those are the correct limits of integration. Sep 3, 2020 at 5:13
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