Use the convolution formula, Theorem 6.9, to obtain the density of \(X+Y\) when \(X\) and \(Y\) are independent and each has the exponential distribution with \(\beta=1\).

Data From Theorem 6.9

Theorem 6.9 Let X and Y be independent and let X have density fx(x) and Y have density fy (y). Then the density of Z = X + Y is given by the convolution formula fx+y(z) = fx(x)fy(z - x) dx for all z The ratio of random variables Z = Y/X has density |x| fy/x(z) = | | x | x(x)fy(xz) dx for all z