Suppose that the random variables X, Y, and Z have the joint probability density function f (x, y, z) = 8xyz for 0 < x < 1, 0 < y < 1, and 0 < z < 1. Determine the following:

(a) P(X <0.5) 

(b) P(X <0.5, Y <0.5)

(c) P(Z <2) 

(d) P(X <0.5 or Z <2)

(e) E(X) 

(f) P(X <0.5 |Y = 0.5)

(g) P(X <0.5, Y <0.5 | Z = 0.8)

(h) Conditional probability distribution of X given that Y = 0 5 and Z = 0 8.

(i) P(X <0.5 |Y = 0.5, Z = 0.8)