Thread: Marginal PDF from Joint PDF?

Hey all! I'm new here and I've scanned some posts but couldn't find anything that could help me out with this problem. Here's what I'm stuck with:

Let X and Y be two random variables with a joint probability density function f given by:
f(x, y) =
1, if −y < x < y,
0, if 0 < y < 1.

Find the marginal probability density functions of X and Y.

I've been going at this for a while now and any help would be appreciated. Thanks in advance!

Originally Posted by Falantar

Hey all! I'm new here and I've scanned some posts but couldn't find anything that could help me out with this problem. Here's what I'm stuck with:

Let X and Y be two random variables with a joint probability density function f given by:
f(x, y) =
1, if −y < x < y,
0, if 0 < y < 1.

Find the marginal probability density functions of X and Y.

I've been going at this for a while now and any help would be appreciated. Thanks in advance!

I'd start by applying the definition for marginal pdf. Do you know it? If so, where are you getting stuck?

Originally Posted by mr fantastic

I'd start by applying the definition for marginal pdf. Do you know it? If so, where are you getting stuck?

I am pretty sure the marginal PDF of X, is the integral of the joint PDF with respect to y, while the marginal PDF of Y is the integral of joint PDF with respect to x. I am stuck because i'm not sure how to go about doing the integration.

Originally Posted by Falantar

I am pretty sure the marginal PDF of X, is the integral of the joint PDF with respect to y, while the marginal PDF of Y is the integral of joint PDF with respect to x. I am stuck because i'm not sure how to go about doing the integration.

Please be more specific. Where exactly are you stuck? What have you tried? Are you studying (or have you studied) calculus of several variables and covered double integrals?

Originally Posted by mr fantastic

Please be more specific. Where exactly are you stuck? What have you tried? Are you studying (or have you studied) calculus of several variables and covered double integrals?

I have not studied multivariable calculus but i have covered double integrals. I have not yet tried anything because I don't know where to begin. Sorry I don't know how I can be any more specific...

You do not need to know double integrals to find a marginal PDF from a joint PDF. To find the marginal PDF of X, integrate the joint PDF with respect to Y, and set your bounds of integration as the range of Y. Similarly, to find the marginal PDF of Y, integrate the joint PDF with respect to X, and set your bounds of integration as the range of X.

Basically:
and
If you are getting stuck on solving the integral, you're going to have to post the joint PDF or at least some of your work.

Originally Posted by Falantar

I am pretty sure the marginal PDF of X, is the integral of the joint PDF with respect to y, while the marginal PDF of Y is the integral of joint PDF with respect to x. I am stuck because i'm not sure how to go about doing the integration.

Start by drawing the region defined by the support. Can you do this?