Thread: Marginal Density

Given:
The joint probability density function of X and Y is:


with bounds .

Question: Find the marginal density of X.
In the solutions key, when they find the marginal density of X they use the limits of integration .

Why do they do this and not use as the limits?

Hello,

Originally Posted by utopiaNow

Given:
The joint probability density function of X and Y is:


with bounds .

Question: Find the marginal density of X.
In the solutions key, when they find the marginal density of X they use the limits of integration .

Why do they do this and not use as the limits?

-y<x<y gives :
. y>x
. or -y<x ---> y>-x
This means that y>|x|
So you have to take as the lower bound.
Since , the lower bound will be |x|.

Is it clearer this way ?

Originally Posted by utopiaNow

Given:
The joint probability density function of X and Y is:


with bounds .

Question: Find the marginal density of X.
In the solutions key, when they find the marginal density of X they use the limits of integration .

Why do they do this and not use as the limits?

Have you tried drawing the region over which the given joint pdf is non-zero? Draw the lines y = x and y = -x. The region is the area above the V-shape, that is the region above the line y = x from x = 0 to x = oo and the region above the line y = -x from x = 0 to x = -oo. This region is alternatively defined by the area above y = |x|.

Thanks guys for the explanations, drawing the region helped me and also understanding that we have to let the lower bound be the max of 0 and |x|.