Page 194 - Applied statistics and probability for engineers

172 Chapter 5/Joint Probability Distributions

(
(

(a) Probability that two bits have high distortion and one has (c) P Y >3) (d) P X < ,Y < 2)
2
moderate distortion (e) E X ( ) (f) E Y ( )
(b) Probability that all three bits have low distortion
(c) Probability distribution, mean, and variance of X (g) Marginal probability distribution of X
(
| (
1)
(d) Conditional probability distribution, conditional mean, and (h) Conditional probability distribution of Y given X = 1
(j) P Y < 2 |
X =
conditional variance of X given that Y = 2 (i) E Y X = ) 1
5-13. Determine the value of c such that the function (k) Conditional probability distribution of X given Y = 2
5-18. The conditional probability distribution of Y given X =
, (
f x y) = cxy for 0 < < 3 and 0 < < 3 satisies the properties is f Y x| ( y) = − xy x
x
y
of a joint probability density function. xe for y > 0, and the marginal probability distri-
Determine the following: bution of X is a continuous uniform distribution over 0 to 10.
(a) Graph f Y X| (
−
y) =
xy
(
(
(a) P X < ,Y <2 3) (b) P X < 2 5) xe for y > 0 for several values of x.
.
Determine:
(
1 (
(c) P <Y < 2 5) (d) P X >1 8 , <Y < 2 5) (b) P Y < 2 | X = 2) (c) E Y X = ) 2
| (
(
.
.
.
1
(
(e) E X ( ) (f) P X < ,Y < 4) (d) E Y X| ( = x) (e) f XY ( x, y) (f) f y ( )

0

Y
(g) Marginal probability distribution of X 5-19. Two methods of measuring surface smoothness are used
(h) Conditional probability distribution of Y given that X = 1 5 to evaluate a paper product. The measurements are recorded
.
| (
(
.
(i) E Y X) = . ) 5 (j) P Y < 2 | X = 1 5) as deviations from the nominal surface smoothness in coded
1
(k) Conditional probability distribution of X given that Y = 2 units. The joint probability distribution of the two measure-
5-14. Determine the value of c that makes the function ments is a uniform distribution over the region 0 < <x 4, 0 < y,
f x, ( y) = ( y) a joint probability density function over the and x −1< < x + 1. That is, f XY ( x, y) = c for x and y in the
c x +
y
x
y
range 0 < <x 3 and x < < + 2. region. Determine the value for c such that f ( x, y) is a joint
XY
Determine the following: probability density function.
(
1 (
(a) P X < ,Y <1 2) (b) P < X < 2) Determine the following:
(
(
(
(
(c) P Y >1) (d) P X < ,Y < 2) (a) P X < 0 5 . ,Y < 0 5) (b) P X < 0 5)
.
.

2
(e) E X ( ) (f) V X ( ) (c) E X ( ) (d) E Y ( )
(g) Marginal probability distribution of X (e) Marginal probability distribution of X
(h) Conditional probability distribution of Y given that X = 1 (f) Conditional probability distribution of Y given X = 1
(
| (
| (
(
.
(i) E Y X = ) 1 (j) P Y > 2 | X = 1) (g) E Y X = ) 1 (h) P Y < 0 5| X = 1)
(k) Conditional probability distribution of X given that Y = 2 5-20. The time between surface inish problems in a gal-
5-15. Determine the value of c that makes the function vanizing process is exponentially distributed with a mean of 40
f x, ( y) = ( y) a joint probability density function over the hours. A single plant operates three galvanizing lines that are
c x +
y
range 0 < <x 3 and 0 < < x. assumed to operate independently.
Determine the following: (a) What is the probability that none of the lines experiences a
1 (
(
(a) P X < ,Y <1 2) (b) P < X < 2) surface inish problem in 40 hours of operation?
(
(c) P Y >1) (d) P X < ,Y < 2) (b) What is the probability that all three lines experience
(

2
a surface inish problem between 20 and 40 hours of
(
(
(e) E X) (f) E Y) operation?
(g) Marginal probability distribution of X (c) Why is the joint probability density function not needed to
(h) Conditional probability distribution of Y given X = 1 answer the previous questions?
| (
(
(i) E Y X = ) 1 (j) P Y > 2 | X = 1) 5-21. A popular clothing manufacturer receives Internet
(k) Conditional probability distribution of X given Y = 2 orders via two different routing systems. The time between
5-16. Determine the value of c that makes the function
(
f x, y) = ce − 2 x− 3 y a joint probability density function over the orders for each routing system in a typical day is known to
range 0 < x and 0 < <y x. be exponentially distributed with a mean of 3.2 minutes. Both
systems operate independently.
Determine the following:
(
1 (
(a) P X < ,Y <1 2) (b) P < X < 2) (a) What is the probability that no orders will be received in a
5-minute period? In a 10-minute period?
(c) P Y >3) (d) P X( < 2 Y , < 2 ) (b) What is the probability that both systems receive two orders
(
(e) E X) (f) E Y) between 10 and 15 minutes after the site is oficially open
(
(
(g) Marginal probability distribution of X for business?
(h) Conditional probability distribution of Y given X = 1 (c) Why is the joint probability distribution not needed to
| (
(i) E Y X = ) 1 answer the previous questions?
(j) Conditional probability distribution of X given Y = 2 5-22. The blade and the bearings are important parts of a lathe.
5-17. Determine the value of c that makes the function The lathe can operate only when both of them work properly. The
(
f x, y) = ce −2 x−3 y , a joint probability density function over the lifetime of the blade is exponentially distributed with the mean
range 0 < x and x < y. three years; the lifetime of the bearings is also exponentially
Determine the following: distributed with the mean four years. Assume that each lifetime
(
1 (
(a) P X < ,Y <1 2) (b) P < X < 2) is independent.