An order of 15 printers contains 4 with a graphics- enhancement feature, 5 with extra memory, and 6 with both features. Four printers are selected at random, without replacement, from this set. Let the random variables X, Y, and Z denote the number of printers in the sample with graphics enhancement only, extra memory only, and both, respectively.

(a) Describe the range of the joint probability distribution of X, Y, and Z.

(b) Is the probability distribution of X, Y, and Z multinomial? Why or why not?

(c) Determine the conditional probability distribution of X given that Y = 2.

Determine the following:

(d) P(X = 1, Y = 2, Z = 1) 

(e) P(X = 1, Y = 1)

(f) E(X ) and V (X ) 

(g) P(X = 1, Y = 2 | Z = 1)

(h) P(X = 2 |Y = 2)

(i) Conditional probability distribution of X given that Y = 0 and Z = 3.