I am looking through Marcel Finan's 'A Probability Course for the Actuaries' and I am stuck on problem 42.2. It is as follows:
Let X be an exponential random variable with parameter $\lambda$ and Y be a uniform random variable on [0,1]. Find the probability density function of X+Y.
I have $f_X(x) = \lambda e^{-\lambda x}, x \ge 1$ and $f_Y(y) = 1, 0 \le y \le 1$.
I am kind of lost as to where to go from there. According to the solution key, the final answer should come out relatively 'nice', but it's in two different pieces - one with $0 \le a \le 1$ and the other with $a \ge 1$. I can see from the integration region why there would be two separate pieces. The real issue isn't with that. I think I'm just setting up my integrals incorrectly.
For $0 \le a \le 1$ I have
$f_{X+Y}(a) = \int_0^1 \lambda e^{-\lambda x}dx$
Any thoughts on where I might be going wrong? Thanks in advance!
Addition of independent random variables corresponds to convolution of distributions so say for $0\leq \alpha \leq 1$ you should have a double integral rather than a single integral (in terms of densities).
Here's a hint to get your started for $0 \leq \alpha \leq 1$:
$X+Y \leq \alpha$ is equivalent to $0 \leq X \leq \alpha-Y$ and $0\leq Y \leq \alpha$.