What is the Marginal PDF of X?

[Phox]

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?

 

[Phox]

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

 

[Ray Vickson]

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.

 

[Phox]

I've graphed it. I'm not sure what this tells me

 

[Ray Vickson]

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.