Question, Joint Distribution Functions

Another question presented in lecture.

Consider a random variable (X,Y) which is uniformly distributed over the triangle T = {(x,y) | x > 0, y > 0, x+y < 9}.

a) Explain why X and Y are dependent (no calculations required).

b) Find f_y(y), the marginal pdf of Y. Check that it is a probability density.

c) For a fixed y ∈ (0,9), write down the set of values of x for which f_x|y=y (x|y) is non-zero.

d) Find f_x|y=y (x|y) over the range that you derived in part d.

e) Calculate Cov(X,Y).

Thanks!

Re: Question, Joint Distribution Functions

Hey Radiance.

Hint for a) If two random variables are independent, then their domains are independent and the result would be that the limits of the integral would just be constants instead of one with dependencies in the limits. Since the limits are dependent, it means that the probabilities are dependent as well (since they represent probabilities).

More formally P(A = a, B = b) = P(A = a)*P(B = b) for independence and the only way this is possible is if Integral over A and B f(a,b)dadb = Integral A f(a)da * Integral B g(b) db.

Re: Question, Joint Distribution Functions

I'm guessing for 2 we just take the double integrals over x and y and show that they are the product of the probabilities.

Re: Question, Joint Distribution Functions

For marginals, you need to integrate out the x variable to get the marginal f_y(y).