Help with a density function

**emonimous** · Feb 13th 2011, 11:50 AM

Question:
Suppose that random variables X and Y have a joint density function given by

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

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

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
$f_{X,Y}(x,y)=f_X(x)\bullet f_Y(y)$ 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.

**emonimous** · Feb 13th 2011, 11:56 AM

Something just hit me, in order to find f(x) do I compute the density function by integrating in respect to dy?

**harish21** · Feb 13th 2011, 02:38 PM

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

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

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

**emonimous** · Feb 13th 2011, 03:13 PM

Originally Posted by harish21

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

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

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

Thanks a lot! It confirms what I said!

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