1. ## covariance and corellation

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)?

2. Originally Posted by thandu3
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?

[snip]
$\rho = \frac{Cov(X, Y)}{\sigma_X \sigma_Y}$ where $Cov(X, Y) = E(XY) - E(X) E(Y)$.

$E(XY) = \int_{x=0}^1 \int_{y=0}^{y=x} (xy) \, 8xy \, dy \, dx$.

$E(X)$, $E(Y)$, $\sigma_X$ and $\sigma_Y$ are calculated from the marginals $f_X(x) = \int_{y = 0}^{y = x} 8xy \, dy$ and $f_Y(y) = \int^{x=1}_{x=y} 8xy \, dx$.

3. Originally Posted by thandu3
[snip]
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) .... ?

4. ## thanks

Thanks a lot for your kind reply.Thanks for the idea.I was working on it but an idea can make it solve quickly.