# Help with a density function

• Feb 13th 2011, 11:50 AM
emonimous
Help with a density function
Question:
Suppose that random variables X and Y have a joint density function given by

$\displaystyle f_{X,Y} (x,y)=\left\{\begin{array}{cc}x+y,&\mbox{ for } 0 \leq x\leq 1, 0\leq y\leq 1\\0, & \mbox{ otherwise } \end{array}\right.$

$\displaystyle a) \mbox {Find the density functions of X and Y,} i.e., f_X(x) \mbox {and} f_Y(y)$
$\displaystyle b)\mbox {Find E[X] and }Var[Y]$

What I have done so far:
Frankly I have been staring at this problem for a while. I'm not sure where to start. I cannot apply the following definition for joint probability functions
$\displaystyle f_{X,Y}(x,y)=f_X(x)\bullet f_Y(y)$ as the probability function is not independent. Anyone care to point me in the right direction? Once I could get it figured out, I can solve for b.

PS. Please don't post any solutions as it defeats the whole purpose of asking for help, I won't learn anything that way.
• Feb 13th 2011, 11:56 AM
emonimous
Something just hit me, in order to find f(x) do I compute the density function by integrating in respect to dy?
• Feb 13th 2011, 02:38 PM
harish21
$\displaystyle f_X(x)=\displaystyle \int_y f_{X,Y}(x,y) dy$

$\displaystyle f_{Y}(y)=\displaystyle \int_x f_{X,Y}(x,y) dx$

After you have the marginal distributions, you can easily find the mean and variances of each.
• Feb 13th 2011, 03:13 PM
emonimous
Quote:

Originally Posted by harish21
$\displaystyle f_X(x)=\displaystyle \int_y f_{X,Y}(x,y) dy$

$\displaystyle f_{Y}(y)=\displaystyle \int_x f_{X,Y}(x,y) dx$

After you have the marginal distributions, you can easily find the mean and variances of each.

Thanks a lot! It confirms what I said!