# Marginals

• Nov 20th 2008, 10:40 AM
Nightfly
Marginals
I need some help with the limits of the integrals when you work out the marginal pdf of X or Y.

I've got the formulae

$\displaystyle f(x)= \int_{-\infty}^{\infty}f(x,y)dy$
for the marginal pdf of X and

$\displaystyle f(y)= \int_{-\infty}^{\infty}f(x,y)dx$
for Y

Then there's the question
Random variables X and Y have joint pdf

$\displaystyle f(x,y)=\left\{\begin{array}{cc}xy,&\mbox{for} (x,y)inA\\0, & \mbox{ elsewhere } \end{array}\right.$

where A is the region bounded between $\displaystyle y=x$ and $\displaystyle y=x^2$

Looking at the solutions, the limits for f(x) are between $\displaystyle x$ and $\displaystyle x^2$ and I think I get that one but for f(y) they're between $\displaystyle y$ and $\displaystyle y^\frac{1}{2}$ which I can't figure out. Can anyone help?
• Nov 20th 2008, 08:56 PM
CaptainBlack
Quote:

Originally Posted by Nightfly
I need some help with the limits of the integrals when you work out the marginal pdf of X or Y.

I've got the formulae

$\displaystyle f(x)= \int_{-\infty}^{\infty}f(x,y)dy$
for the marginal pdf of X and

$\displaystyle f(y)= \int_{-\infty}^{\infty}f(x,y)dx$
for Y

Then there's the question
Random variables X and Y have joint pdf

$\displaystyle f(x,y)=\left\{\begin{array}{cc}xy,&\mbox{for} (x,y)inA\\0, & \mbox{ elsewhere } \end{array}\right.$

where A is the region bounded between $\displaystyle y=x$ and $\displaystyle y=x^2$

Looking at the solutions, the limits for f(x) are between $\displaystyle x$ and $\displaystyle x^2$ and I think I get that one but for f(y) they're between $\displaystyle y$ and $\displaystyle y^\frac{1}{2}$ which I can't figure out. Can anyone help?

The marginal pdf for y is:

$\displaystyle f(y)= \int_A f(x,y)dx= \int_{x=y}^{\sqrt{y}} xy \ dx= \left[ \frac{x^2y}{2}\right]_{x=y}^{\sqrt{y}},\ \ y \in (0,1)$

CB