# Probability density function

## Homework Statement

Hello! I'm trying to understand how to solve the following type of problems.

1) Random variables x and y are independent and uniformly distributed on the interval [0; a]. Find probability density function of a random variable z=x-y.

2) Exponentially distributed (p=exp(-x), x>=0) random variables x and y are independent. Find probability density function of a random variable z=x-y.

## Homework Equations

Can someone please check if my attempt to solve the problems is successful or not? I'd appreciate any help :)

## The Attempt at a Solution

(Attached file)

#### Attachments

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Just a comment:
From the exercise I read they want you to find the probability distribution itself, not the accumulated function. A slightly different beast.

And a question:
Do you know about convolution ? (you are more or less working it out on your own here). Understanding that concept makes things a lot easier.

andrewkirk
For part (1) your answer is correct for $z\leq 0$ but not for $0<z<a$. If you substitute $z=0$ into your formula for that latter case you get 0, whereas it should be 1/2. I think the problem will be with the limits used in the inner of your double integrals. The probability should move smoothly from 1/2 to 1 as $z$ goes from 0 to $a$.