# Probability Density of Sum of Random Variables

snipez90

## Homework Statement

Suppose X and Y are independent random variables with X following a uniform distribution on (0,1) and Y exponentially distributed with parameter $\lambda = 1$. Find the density for Z = X + Y. Sketch the density and verify it integrates to 1.

## Homework Equations

If Z = X + Y, and X and Y are independent,

$$f_z(z) = \int_{-\infty}^{\infty}f_Y(z-x)f_X(x)\,dx$$

## The Attempt at a Solution

I am trying to catch up on stats. I think I screwed up somewhere in this problem:

If X ~ Unif(0,1), then $f_X(x)$ should just be 1 on (0,1) right? Also if the parameter of the exponential distribution is 1, then $f_Y(z-x) = e^{-(z-x)}?$

But then the expression for $f_z(z)$ doesn't seem to make much sense. Note I have no idea what I am doing.