I managed to solve this uniform distribution problem and now i am having difilculties with calculating area on graph

Question

So the task says: There are 2 variables that get random variables from 0 to 1. What is the distribution function for z that represents distance between the two variables.

Lets say first variable is x and second is y, z = |x - y|, we can draw a graph like this https://i.stack.imgur.com/xRUe1.jpg , however I really don't know how can i calculat this G area, can someone help me?

Edit: official solution to the problem is $2z - z^2$

Answer 1

OOh of course, i have found the solution it is the area of square which is 1 - area of two triangles with one side of the triangle equal to 1 - z. so the solution would be 1 - (1-z)(1-z) = 2z -z^2

Answer 2

Actually, part of your problem may be, to quote an educational source:

Theorem: The difference of two independent standard uniform random variables has the standard trianglular distribution.

As such, now you known what the answer looks like, so continue with your suggested path or use the provided reference.