What is the Marginal PDF of X?

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Homework Help Overview

The discussion revolves around finding the marginal probability density function (pdf) of the random variable X, given the joint pdf of X and Y as f(x,y) = 8xy for the region defined by 0 ≤ x ≤ y ≤ 1. Participants are attempting to understand the correct bounds for integration in order to derive the marginal pdf of X.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants are discussing the integration process needed to find the marginal pdfs, with some questioning the initial bounds used for integration. There is an exploration of the correct limits for the integrals based on the defined region in the (x,y) plane.

Discussion Status

Some participants have provided insights into the correct bounds for the integrals, suggesting that the bounds for f1(x) should be from x to 1 and for f2(y) from 0 to y. There is an ongoing exploration of how to interpret the graphical representation of the region where the joint pdf is positive.

Contextual Notes

Participants are grappling with the implications of the joint pdf's constraints and how they affect the integration bounds. The need to visualize the region where the joint pdf is greater than zero has been emphasized as a crucial step in understanding the problem.

Phox
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Homework Statement



Let X and Y be random variables of the continuous type having the join p.d.f.:

f(x,y) = 8xy, 0<=x<=y<=1

Find the marginal pdf's of X. Write your answer in terms of x.

Find the marginal pdf's of X. Write your answer in terms of x.

Homework Equations





The Attempt at a Solution



f1(x) = integral(8xy)dy from 0 to 1

f2(y) = integral(8xy)dx from 0 to 1

f1(x) = 4x
f2(x) = 4y

This isn't right. what am I doing wrong?
 
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Ok, so i guess the bounds of f1(x) were supposed to be from x to 1.

And the bounds from f2(y) were supposed to be from 0 to y.

But I don't don't understand why
 
Phox said:

Homework Statement



Let X and Y be random variables of the continuous type having the join p.d.f.:

f(x,y) = 8xy, 0<=x<=y<=1

Find the marginal pdf's of X. Write your answer in terms of x.

Find the marginal pdf's of X. Write your answer in terms of x.

Homework Equations





The Attempt at a Solution



f1(x) = integral(8xy)dy from 0 to 1

f2(y) = integral(8xy)dx from 0 to 1

f1(x) = 4x
f2(x) = 4y

This isn't right. what am I doing wrong?

Before doing any calculations, draw the region f > 0 in the (x,y) plane; that is, draw the region
0 ≤ x ≤ y ≤ 1.
 
I've graphed it. I'm not sure what this tells me
 
Phox said:
I've graphed it. I'm not sure what this tells me

The marginal pdf ##f_X(x)## of X is the y-integral (with fixed x), integrated over the whole relevant y-region for that value of x. The drawing tells you what that region that would be.
 

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