Assume $ X, Y$ are two random variables that are exponential distributed with parameter $ \beta > 0$ and independet from each other. Define $ Z := X + Y$ . Determine the density function of $ Z$ .

Since $ X$ and $ Y$ are i.i.d., we are allowed to use the fact that X + Y possesses the density function $ $ \int_{-\infty}^{\infty} f(r-s)f(s) ds,$ $

with $ f(r-s)$ respectively $ f(s)$ being the density function of $ X$ and $ Y$ . For $ s \ge 0$ , the density function of an exponential distributed random variable is given by $ f(s) = \beta e^{-\beta s}$ , and since it is $ 0$ otherwise, we can rewrite the integral:

$ $ \int_{0}^{\infty} f(r-s)f(s) ds = \int_{0}^{\infty} \beta e^{-\beta (r-s)} \beta e^{-\beta s} ds = \beta^2\int_{0}^{\infty} e^{-r\beta} ds,$ $

which disappears to $ \infty$ , unfortunately. What’s my mistake here?