Finding Probability Density Functions for Independent Random Variables

In summary, the conversation discusses finding the probability density function of a random variable z=x-y for two different scenarios. The first scenario involves independent and uniformly distributed random variables x and y, while the second scenario involves exponentially distributed random variables x and y. The person asking for help also mentions their attempt at solving the problems and asks for feedback. One commenter suggests understanding the concept of convolution, while another points out an error in the provided solution for the first scenario.
  • #1
proton4ik
15
0

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]
 

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  • #2
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
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.
 

Related to Finding Probability Density Functions for Independent Random Variables

What is a probability density function (PDF)?

A probability density function (PDF) is a mathematical function that describes the probability of a random variable falling within a particular range of values. It is typically used to analyze continuous data and is represented by a curve on a graph.

How is a PDF different from a probability distribution function (PDF)?

While both a probability density function and a probability distribution function represent the probabilities of a random variable, the main difference is that a PDF is used for continuous data while a PDF is used for discrete data. A PDF can be thought of as the "smoothed out" version of a PDF.

How do you calculate the area under a PDF curve?

To calculate the area under a PDF curve, you can use the integral function. The area under the curve represents the probability of the random variable falling within a specific range of values.

What is the relationship between a PDF and a cumulative distribution function (CDF)?

A cumulative distribution function (CDF) is the integral of a PDF and represents the probability of a random variable being less than or equal to a certain value. In other words, the CDF is the "accumulation" of probabilities from the PDF.

Can a PDF be used to find the probability of a specific value?

No, a PDF cannot be used to find the probability of a specific value. Instead, it is used to find the probability of a range of values. The probability of a specific value in a PDF is equal to zero, as the area under a single point is infinitesimally small.

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