Random variables $X$ and $Y$ follow a joint distribution $$f(x, y) = \left\{ \begin{array}{ll} 2,& 0 < x \leq y < 1,\\ 0,&\text{otherwise}\end{array}\right. $$
Determine the correlation coefficient between $X$ and $Y$ .
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$\begingroup$We need to calculate cov(X,Y)=EXY - EX*EY, var(X) and var(Y). To do it, we have to know marginal distributions of both random variables X and Y. This can be done by "integrating the other variable out" of the joint density function. I will show you how to calculate the marginal density function of X:
$ f(x)= \int^{1}_{x} f(x,y) dy = \int^{1}_{x} 2 dy = 2(1-x)$ for $x \in (0,1)$.
Are you able to continue?
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