Suppose that X; Y have constant joint density on the triangle with corners at (4; 0), (0; 4), and the origin. a) Find P(X < 3; Y < 3). b) Are X and Y independent

The triangle (call it T ) has base and height 4, so its area is 1/2*4*4 = 8. Then the joint density function is

where T is the set

(a) I've attached an image of the integration region.

(b) X and Y are independent if the joint distribution is equal to the product of their marginal distributions.

Get the marginal distributions of one random variable by integrating the joint density over all values of the other variable:

Clearly, , so they are not independent.

(1,4) is on graph of f(x)

f(1) = 4

so 1/4 f(x)

1/4 f(1)

1/4. ×4

1

so point for function 1/4. f(x) is (1,1)