# 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)

Thank you in advance[/B]

#### Attachments

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## Answers and Replies

BvU
Science Advisor
Homework Helper
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
Science Advisor
Homework Helper
Gold Member
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##.

EDIT: Just saw BvU's answer and I agree with that. Your answer is a CDF but a PDF has been requested. You can get the PDF by differentiating the CDF.