Determine the value of c that makes f_XYZ (x, y, z) = c a joint probability density function over the region x > 0, y > 0, z > 0, and x + y + z < 1.

Determine the following:

(a) P(X <0.5, Y <0.5, Z <0.5) 

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

(c) P(X <0.5) 

(d) E(X)

(e) Marginal distribution of X

(f) Joint distribution of X and Y

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

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