Statistics Problem. Joint Probability?

Question

I have been working on this problem and am having trouble getting started. Can anyone help?

Let X and Y have the joint probability density function given by f(x,y) = 2 for 0 <= x <= 1, 0 <= y <= 1, and 0 <= x+y <= 1. Find
a.) P(X >= 1/2 | Y <= 1/4)
b.) P(X >= 1/2 | Y = 1/4)

thanks!

Answer 1

a) P(X >= 1/2 | Y <= 1/4) = P(X >= 1/2 and Y <= 1/4) / P(Y <= 1/4)
= ∫(0 to 1/4) ∫(1/2 to 1-y) 2 dx dy / ∫(0 to 1/4) ∫(0 to 1-y) 2 dx dy
= ∫(0 to 1/4) (1 - 2y) dy / ∫(0 to 1/4) (2 - 2y) dy
= (1/4 - 1/16) / (1/2 - 1/16)
= 3/7.

b) Restrict the domain to the line y = 1/4, 0 <= x <= 3/4. Rescale f(x, y) by a factor k to get a 1-dimensional pdf for x:
g(x) = k f(x, 1/4) = 2k
∫(0 to 3/4) g(x) dx = 1 => (3/4) (2k) = 1 => k = 2/3.
So g(x) = 4/3 and P(X >= 1/2 | Y = 1/4) = ∫(1/2 to 3/4) (4/3) dx = (4/3) (3/4 - 1/2) = 1/3.

You should be able to convince yourself of this geometrically, as well.