# Thread: Calculating distribution function of Z = X + Y with X & Y continuous r.v.s

1. ## Calculating distribution function of Z = X + Y with X & Y continuous r.v.s

Let X & Y be continuous random variables with joint probability density function $\displaystyle f_{X,Y}(x,y)$. If Z is defined by Z = X + Y, write the distribution function of Z, that is $\displaystyle F_Z(z) := \mathbb{P}[X + Y \leq z]$ as an iterated integral, where the integration is done first with respect to x.

This is a sample question for a test I have coming up, not sure where to go with it, any help appreciated!

2. p(X + Y < z) = p(X < z - Y)

3. Had seen that but not sure how to use that fact

4. Integral limits?

5. are they both uniform (0,1) random variables.
If so, this is a classic question.
In tht case you will need to break the region into two parts, from 0 to 1 and from 1 to 2.
Then you can use geometry.
Otherwise.........

$\displaystyle P(X+Y\le a) =\int_0^a\int_0^{a-x}f(x,y)dydx =\int_0^a\int_0^{a-y}f(x,y)dxdy$