Stat Joint Distribution Question

Question

Given the continuous random variables X and Y that have a joint distribution:

fxy(x,y) = 4xy where 0 < x < 1 and 0 <y < 1

a) are X and Y independent

b) what is the correlation between X and Y

Answer 1

If we can write f(x,y) = f(x)f(y) and the x and y domains are independent of one another, then x and y are independent.

In this case, as f(x,y) = 4xy, we can write f(x) = 2x and f(y) = 2y.

The function is defined on [0, 1] x [0, 1], so the x and y domains are also independent.

Thus, x and y are independent.

When x and y are independent, the correlation is 0.