# Convolution with Uniform and Exponential Random Variables

If $X$ Unif~$[2, 5]$ and $Y$ Exp~$(4)$ are independent, what is the probability density function of $X + Y$ ?

I'm a bit confused about what the limits of integration should be to find the convolution:

We let $Z = X + Y$. Then,

$f_Z (z) = \int f_X (x) f_Y (z-x) \ dx$. I know the individual density functions, but I'm unsure of the limits of integration.

## 1 Answer

Assuming $z \geq 2$, we need $x$ between 2 and 5. Also, we need $z-x \geq 0$ which is equivalent to $x \leq z$.

So lower limit is 2, upper limit is $\min(5,z).$

• Thank you! Just to clarify--we need x between 2 and 5 since that is where the density is non zero? Similarly for the other density?
– user288829
May 7 '16 at 23:56
• yup, if you go beyond that, that is fine too, just that the density value at that region is zero. May 8 '16 at 3:54