Joint PDF

panda* · May 6th 2009, 04:35 AM

The joint probability distribution is given by,

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

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

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

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

**mr fantastic** · May 6th 2009, 04:56 AM

Originally Posted by panda*

The joint probability distribution is given by,

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

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

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

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

I will get you started:

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

Show that $\int_{0}^{1} f_X(x) \, dx = 1$ . Alternatively, show that $\int_{0}^{1} f_Y(y) \, dx = 1$ .

**matheagle** · May 6th 2009, 11:17 PM

This uniform on half of the unit square has been posted here a few times.

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