Question

Obtain the correlation coefficient for the random variables X and Y

having joint density f(x,y) = 6x for 0 < x ⋜ y < 1.

Answer 1

To calculate the correlation coefficient for the random variables X and Y, first find the means and variances of X and Y. Then calculate the covariance of X and Y and divide it by the product of their standard deviations.

Step-by-step explanation:

The correlation coefficient measures the strength of the linear association between the random variables X and Y. To calculate the correlation coefficient, we need to first find the means and variances of X and Y. In this case, X has a mean of 5 and a variance of 6, while Y has a mean of 2 and a variance of 1. Next, we need to find the covariance of X and Y, which is given by the equation:

cov(X,Y) = E[(X-μx)(Y-μy)]

Finally, the correlation coefficient, r, is calculated by dividing the covariance by the product of the standard deviations of X and Y:

r = cov(X,Y) / (σx * σy)

Plugging in the values, we find that r = 0.5.