1) Suppose that two random variables X & Y are jointly distributed according to the pdf f(x,y)=8xy , 0<=y<=x<=1. What is the correlation coefficient of X & Y?
2) Consider the sample space S={(-2,4),(-1,1),(0,0),(1,1),(2,4)}.Where each point are assumed to be equally likely.Define the random variable X to be the first component of a sample point and Y to be the second.Then X(-2,4)=-2, Y(-2,4)=4 and so on.Is X and Y independent? What is the covariance Cov(X,Y)?
I suggest that you calculate the covariance first: Cov(X, Y) = E(XY) - E(X) E(Y).
If it's not equal to zero then you know that X and Y are not independent.
However ..... if it is equal to zero then it's still unknown whether X and Y are independent or not. In which case you should consider doing a couple of simple calculations eg. Does Pr(Y = 1) equal Pr(Y = 1 | X = 0) .... ?