Suppose that the random variables X, Y, and Z have the joint probability density function f_XYZ (x, y, z) = c over the cylinder x² + y² < 4 and 0 < z < 4. Determine the constant c so that f_XYZ (x, y, z) is a probability density function.

Determine the following:

(a) P(X² + Y² <2) 

(b) P(Z <2)

(c) E(X) (d) P(X <1|Y = 1)

(e) P(X2 + Y 2 <1| Z = 1)

(f) Conditional probability distribution of Z given that X = 1 and Y = 1.