Correlation between random variables

[Evo8]

Homework Statement

Find correlation between random variables x and y in the following:
$$P_{x,y}(x,y)=A \ xy \ e^{-(x^2)}e^{-\frac{y^2}{2}}u(x)u(y)$$

Homework Equations

The co-variance ##\sigma_{xy}=\overline{(x-\bar{x})(y-\bar{y})}## or ##\sigma_{xy}=\overline{xy}-\bar{x}\bar{y}##

The concept of co variance is a natural extension of the concept of variance. Definition -> ##\sigma_{x}^{2}=\overline{(x-\bar{x})(x-\bar{x})}##

Variables x and y are uncorrelated ##(\sigma_{xy}=0)## if ##\overline{xy}=\bar{x}\bar{y}##

Correlation coefficient ##\rho_{xy}=\frac{\sigma_{xy}}{\sigma{x}\sigma{y}}## if x and y are uncorrelated then ##\rho_{xy}=0##

The Attempt at a Solution

Ive solved for A in a previous problem with the same values and came up with ##A=\frac{2}{3}##

Im a little confused as to how I go about finding the "correlation" between x and y in this problem. I've posted a few points from my text that seem like they are relevant or helpful. Am I basically trying to find the correlation coefficient ##\rho_{xy}## and that will give me the correlation? From the definition above I need to know ##\sigma{xy} \ and \ \sigma{x} \ and \ sigma{y}##

I don't really under stand how to find the variance i guess? Even with the definition above. How do I incorporate the bars? Averages?

Any help is appreciated!

 

[BruceW]

yeah. If you have the explicit equation for P(x,y) and you know what are the possible values x and y can take, then you can calculate the quantities you have listed in your relevant section. As you say, the main thing is to use the correct equation for the average of a quantity.

hint: it is slightly more complicated than the 1 variable case, because you have a pdf which depends on two random variables x and y. But hopefully you remember the equation for the average in this case? (Or have it written down somewhere?)