Let X and Y have joint probability density function fx y 2

Let X and Y have joint probability density function f(x, y) = 2 I(0, 1)(x)I (0, 1)(y)I(0, 1)(x + y) (a) Compute the marginal densities fx(x) and fy(y). Are X and Y independent?

Solution

f(x)= integration from (0 to 1-x) [ 2dy] = [2(1-x)]....
f(y)= integration from (0 to 1-y) [ 2dx] = [2(1-y)]....

f(x)*f(y) = [2(1-x)]* [2(1-y)] which is not equal to f(x,y)...so, they ar enot independent...