Joint PDF

• May 6th 2009, 04:35 AM
panda*
Joint PDF
The joint probability distribution is given by,

$\displaystyle f_{(X,Y)}(x,y) = 2$ if $\displaystyle 0 \leq x \leq y \leq 1$, 0 otherwise.

(i) Determine the marginal density functions of X and Y. Check that it is indeed a pdf.

(ii) Calculate the means $\displaystyle \mathbb{E}X$ and VarX.

(iii) Find the conditional density of X given Y = y for $\displaystyle 0 \leq y \leq 1$
• May 6th 2009, 04:56 AM
mr fantastic
Quote:

Originally Posted by panda*
The joint probability distribution is given by,

$\displaystyle f_{(X,Y)}(x,y) = 2$ if $\displaystyle 0 \leq x \leq y \leq 1$, 0 otherwise.

(i) Determine the marginal density functions of X and Y. Check that it is indeed a pdf.

(ii) Calculate the means $\displaystyle \mathbb{E}X$ and VarX.

(iii) Find the conditional density of X given Y = y for $\displaystyle 0 \leq y \leq 1$

I will get you started:

(i) $\displaystyle f_X(x) = \int_{y=x}^{y=1} 2 \, dy$ and $\displaystyle f_Y(y) = \int_{x=0}^{x=y} 2 \, dx$.

Show that $\displaystyle \int_{0}^{1} f_X(x) \, dx = 1$. Alternatively, show that $\displaystyle \int_{0}^{1} f_Y(y) \, dx = 1$.
• May 6th 2009, 11:17 PM
matheagle
This uniform on half of the unit square has been posted here a few times.