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Probability density function

  • #1
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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

Answers and Replies

  • #2
BvU
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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.
 
  • #3
andrewkirk
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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.
 

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