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150 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

and the binomial distributions for X and Y can be used to determine these probabilities as
P1X 12 0.9639 and P1Y 12 0.9831 . Therefore, P1X 1, Y 12 0.948 .
Consequently, the probability that the shipment is accepted for use in manufacturing is
0.948 even if 1% of the parts do not conform to specifications. If the supplier and the pur-
chaser change their policies so that the shipment is acceptable only if zero nonconforming
parts are found in the sample, the probability that the shipment is accepted for production is
still quite high. That is,

P1X 0, Y 02 P1X 02P1Y 02 0.605

This example shows that inspection is not an effective means of achieving quality.

EXERCISES FOR SECTION 5-1
5-1. Show that the following function satisfies the proper- (b) The conditional probability distribution of Y given that
ties of a joint probability mass function. X 1.
(c) The conditional probability distribution of X given that
Y 2.
x y f XY (x, y)
(d) E1Y 0 X 12
1 1 1 4
(e) Are X and Y independent?
1.5 2 1 8
5-9. Show that the following function satisfies the proper-
1.5 3 1 4 ties of a joint probability mass function.
2.5 4 1 4
3 5 1 8
x y f XY (x, y)
1 2 1 8
5-2. Continuation of Exercise 5-1. Determine the following
0.5 1 1 4
probabilities:
0.5 1 1 2
(a) P1X 2.5, Y 32 (b) P1X 2.52
1 2 1 8
(c) P1Y 32 (d) P1X
1.8, Y
4.72
5-3. Continuation of Exercise 5-1. Determine E1X 2 and E1Y 2.
5-4. Continuation of Exercise 5-1. Determine 5-10. Continuation of Exercise 5-9. Determine the follow-
(a) The marginal probability distribution of the random ing probabilities:
variable X. (a) P1X 0.5, Y 1.52 (b) P1X 0.52
(b) The conditional probability distribution of Y given that (c) P1Y 1.52 (d) P1X
0.25, Y 4.52
X 1.5. 5-11. Continuation of Exercise 5-9. Determine E(X) and
(c) The conditional probability distribution of X given that E(Y).
Y 2.
5-12. Continuation of Exercise 5-9. Determine
(d) E1Y 0 X 1.52
(a) The marginal probability distribution of the random
(e) Are X and Y independent?
variable X.
5-5. Determine the value of c that makes the function (b) The conditional probability distribution of Y given that
f 1x, y2 c 1x y2 a joint probability mass function over the X 1.
nine points with x 1, 2, 3 and y 1, 2, 3. (c) The conditional probability distribution of X given that
5-6. Continuation of Exercise 5-5. Determine the following Y 1.
probabilities: (d) E1X 0 y 12
(a) P1X 1, Y 42 (b) P1X 12 (e) Are X and Y independent?
(c) P1Y 22 (d) P1X 2, Y 22
5-13. Four electronic printers are selected from a large lot
5-7. Continuation of Exercise 5-5. Determine E1X 2, E1Y 2, of damaged printers. Each printer is inspected and classified
V1X 2, and V1Y 2. as containing either a major or a minor defect. Let the random
5-8. Continuation of Exercise 5-5. Determine variables X and Y denote the number of printers with major
(a) The marginal probability distribution of the random and minor defects, respectively. Determine the range of the
variable X. joint probability distribution of X and Y.