1. ## Marginal Density

Given:
The joint probability density function of X and Y is:
$
f(x, y) = c(y^2 - x^2)e^{-y}
$

with bounds $-y < x < y, 0 < y < \infty$.

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 $|x|\ and\ \infty$.

Why do they do this and not use $0\ and\ \infty$ as the limits?

2. Hello,
Originally Posted by utopiaNow
Given:
The joint probability density function of X and Y is:
$
f(x, y) = c(y^2 - x^2)e^{-y}
$

with bounds $-y < x < y, 0 < y < \infty$.

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 $|x|\ and\ \infty$.

Why do they do this and not use $0\ and\ \infty$ as the limits?
-y<x<y gives :
. y>x
. or -y<x ---> y>-x
This means that y>|x|
So you have to take $\max\{0,|x|\}$ as the lower bound.
Since $|x|\geq 0$, the lower bound will be |x|.

Is it clearer this way ?

3. Originally Posted by utopiaNow
Given:
The joint probability density function of X and Y is:
$
f(x, y) = c(y^2 - x^2)e^{-y}
$

with bounds $-y < x < y, 0 < y < \infty$.

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 $|x|\ and\ \infty$.

Why do they do this and not use $0\ and\ \infty$ 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|.

4. 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|.