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166 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

5-36. Continuation of Exercise 5-34. Determine the 5-46. Determine the value of c that makes the function
following: f 1x, y2 ce 2x 3y a joint probability density function over
(a) Marginal probability distribution of the random variable X the range 0 x and x y.
(b) Conditional probability distribution of Y given that X 1.5 5-47. Continuation of Exercise 5-46. Determine the
(c) E1Y ƒ X 2 1.52 following:
(d) P1Y 2 ƒ X 1.52 (a) P1X 1, Y 22 (b) P11 X 22
(e) Conditional probability distribution of X given that Y 2 (c) P1Y
32 (d) P1X 2, Y 22
5-37. Determine the value of c that makes the function (e) E1X 2 (f) E1Y 2
f(x, y) c(x y) a joint probability density function over the 5-48. Continuation of Exercise 5-46. Determine the
range 0 x 3 and x y x 2. following:
5-38. Continuation of Exercise 5-37. Determine the (a) Marginal probability distribution of X
following: (b) Conditional probability distribution of Y given X 1
(a) P1X 1, Y 22 (b) P11 X 22 (c) E1Y ƒ X 12
(c) P1Y
12 (d) P1X 2, Y 22 (d) P1Y 2 ƒ X 12
(e) E(X) (e) Conditional probability distribution of X given Y 2
5-39. Continuation of Exercise 5-37. Determine the 5-49. Two methods of measuring surface smoothness are
following: used to evaluate a paper product. The measurements are
(a) Marginal probability distribution of X recorded as deviations from the nominal surface smoothness
(b) Conditional probability distribution of Y given that X 1 in coded units. The joint probability distribution of the
(c) E1Y ƒ X 12 two measurements is a uniform distribution over the re-
gion 0 x 4, 0 y, and x 1 y x 1. That is,
(d) P1Y
2 ƒ X 12
(e) Conditional probability distribution of X given that f XY (x, y) c for x and y in the region. Determine the value for
Y 2 c such that f XY (x, y) is a joint probability density function.
5-40. Determine the value of c that makes the function 5-50. Continuation of Exercise 5-49. Determine the
f(x, y) cxy a joint probability density function over the range following:
0 x 3 and 0 y x. (a) P1X 0.5, Y 0.52 (b) P1X 0.52
5-41. Continuation of Exercise 5-40. Determine the (c) E1X 2 (d) E1Y 2
following: 5-51. Continuation of Exercise 5-49. Determine the follow-
ing:
(a) P1X 1, Y 22 (b) P11 X 22
(a) Marginal probability distribution of X
(c) P1Y
12 (d) P1X 2, Y 22 (b) Conditional probability distribution of Y given X 1
(e) E(X) (f) E(Y)
(c) E1Y ƒ X 12
5-42. Continuation of Exercise 5-40. Determine the (d) P1Y 0.5 ƒ X 12
following:
(a) Marginal probability distribution of X 5-52. The time between surface finish problems in a galva-
(b) Conditional probability distribution of Y given X 1 nizing process is exponentially distributed with a mean of
40 hours. A single plant operates three galvanizing lines that
(c) E1Y ƒ X 12
are assumed to operate independently.
(d) P1Y
2 ƒ X 12 (a) What is the probability that none of the lines experiences
(e) Conditional probability distribution of X given Y 2
a surface finish problem in 40 hours of operation?
5-43. Determine the value of c that makes the function (b) What is the probability that all three lines experience a sur-
f 1x, y2 ce 2x 3y a joint probability density function over face finish problem between 20 and 40 hours of operation?
the range 0 x and 0 y x. (c) Why is the joint probability density function not needed to
5-44. Continuation of Exercise 5-43. Determine the answer the previous questions?
following: 5-53. A popular clothing manufacturer receives Internet
(a) P1X 1, Y 22 (b) P11 X 22 orders via two different routing systems. The time between
(c) P1Y
32 (d) P1X 2, Y 22 orders for each routing system in a typical day is known to be
(e) E(X) (f) E(Y) exponentially distributed with a mean of 3.2 minutes. Both
5-45. Continuation of Exercise 5-43. Determine the systems operate independently.
following: (a) What is the probability that no orders will be received in a
(a) Marginal probability distribution of X 5 minute period? In a 10 minute period?
(b) Conditional probability distribution of Y given X 1 (b) What is the probability that both systems receive two
(c) E1Y ƒ X 12 orders between 10 and 15 minutes after the site is offi-
(d) Conditional probability distribution of X given Y 2 cially open for business?