Page 195 - Applied statistics and probability for engineers

Section 5-1/Two or More Random Variables 173

(a) What is the probability that the lathe will operate for at least standard deviation of 0.03 gram. Any lamp with less than 1.14
ive years? grams of luminescent ink fails to meet customers’ speciica-
(b) The lifetime of the lathe exceeds what time with 95% tions. A random sample of 25 lamps is collected and the mass
probability? of luminescent ink on each is measured.
5-23. Suppose that the random variables X, Y , and Z (a) What is the probability that at least one lamp fails to meet
(
have the joint probability density function f x, y,z) = 8 xyz for speciications?
0 < < 1, 0 < < 1, and 0 < < 1. Determine the following: (b) What is the probability that ive or fewer lamps fail to meet
x
z
y
(
(
(a) P X < 0 5. ) (b) P X < 0 5 ,Y < 0 5) speciications?
.
.
(
(
(c) P Z < 2) (d) P X < 0 5 or Z < 2) (c) What is the probability that all lamps conform to
.
(
speciications?
.
.
(e) E X ( ) (f) P X < 0 5| Y = 0 5) (d) Why is the joint probability distribution of the 25 lamps not
(
.
.
.
(g) P X < 0 5 ,Y < 0 5| Z = 0 8) needed to answer the previous questions?
.
(h) Conditional probability distribution of X given that Y = 0 5 5-29. The lengths of the minor and major axes are used
.
and Z = 0 8 to summarize dust particles that are approximately ellipti-
(
.
(i) P X < 0 5| Y = 0 5 ,Z = 0 8) cal in shape. Let X and Y denote the lengths of the minor
.
.
5-24. Suppose that the random variables X, Y , and Z have and major axes (in micrometers), respectively. Suppose
the joint probability density function f XYZ ( x, y z) = c over the that f x( ) exp(= − x), < x and the conditional distribu-
,
0
X
cylinder x + y < 4 and 0 < <z 4. Determine the constant c so tion f Y x| ( y = exp[ −( y x , < y. Answer or determine the
2
2
−
)
)
]
x
that f ( x, y z) is a probability density function. following:
,
XYZ
Determine the following: (a) That f Y x| ( y) is a probability density function for any value
(
(
(a) P X + Y < 2) (b) P Z < 2) of x.
2
2
(
(c) E X ( ) (d) P X < Y = 1) (b) P X( < Y) and comment on the magnitudes of X and Y .
|
1
(
,
(e) P X + Y <1| Z = 1) (c) Joint probability density function f XY ( x y).
2
2
(d) Conditional probability density function of X given Y =
(f) Conditional probability distribution of Z given that X = 1 (e) P Y( < |2 X = 1 ) (f) E Y X = 1 ) y.
|
(
and Y = 1.
5-25. Determine the value of c that makes f XYZ ( x, y z) = c a (g) P X( < ,1 Y < 1 ) (h) P Y < 2 )
(
,
(i) c such that P Y( <
0
c) = .9
joint probability density function over the region x > 0, y > 0, (j) Are X and Y independent?
y
z > 0, and x + + <1. 5-30. An article in Health Economics [“Estimation of the
z
Determine the following:
(
(
(a) P X < 0 5 . ,Y < 0 5 ,Z < 0 5) (b) P X < 0 5 ,Y < 0 5) Transition Matrix of a Discrete-Time Markov Chain” (2002,
.
.
.
.
(
(c) P X < 0 5) (d) E X ( ) Vol.11, pp. 33–42)] considered the changes in CD4 white
.
blood cell counts from one month to the next. The CD4 count
(e) Marginal distribution of X is an important clinical measure to determine the severity of
(f) Joint distribution of X and Y HIV infections. The CD4 count was grouped into three distinct
.
(g) Conditional probability distribution of X given that Y = 0 5 categories: 0–49, 50–74, and ≥ 75. Let X and Y denote the
and Z = 0 5. (category minimum) CD4 count at a month and the following
(h) Conditional probability distribution of X given that Y = 0 5 month, respectively. The conditional probabilities for Y given
.
5-26. The yield in pounds from a day’s production is nor- values for X were provided by a transition probability matrix
mally distributed with a mean of 1500 pounds and standard shown in the following table.
deviation of 100 pounds. Assume that the yields on different X Y
days are independent random variables.
(a) What is the probability that the production yield exceeds 0 50 75
1400 pounds on each of ive days next week? 0 0.9819 0.0122 0.0059
(b) What is the probability that the production yield exceeds 50 0.1766 0.7517 0.0717
1400 pounds on at least four of the ive days next week? 75 0.0237 0.0933 0.8830
5-27. The weights of adobe bricks used for construction This table is interpreted as follows. For example, P Y( = 50 | X = 75 )
are normally distributed with a mean of 3 pounds and a stand- = .0717. Suppose also that the probability distribution for X is
0
ard deviation of 0.25 pound. Assume that the weights of the P X = 75 ) = . , P X = 50 ) = . , P X = 0 ) = .02. Determine
(
0
8
0
0
0
(
(
9
bricks are independent and that a random sample of 20 bricks the following:
Y
is selected. (a) P Y( ≤ 50 | X = 50 ) (b) P X = , = 75 )
0
(
(a) What is the probability that all the bricks in the sample (c) E Y X( | = 50 ) (d) f y) (e) f XY ( x y),
(
Y
exceed 2.75 pounds? (f) Are X and Y independent?
(b) What is the probability that the heaviest brick in the sample 5-31. An article in Clinical Infectious Diseases [“Strength-
exceeds 3.75 pounds? ening the Supply of Routinely Administered Vaccines in the
5-28. A manufacturer of electroluminescent lamps knows United States: Problems and Proposed Solutions” (2006,
that the amount of luminescent ink deposited on one of its Vol.42(3), pp. S97–S103)] reported that recommended vac-
products is normally distributed with a mean of 1.2 grams and a cines for infants and children were periodically unavailable or