Page 205 - Applied statistics and probability for engineers

Section 5-3/Common Joint Distributions 183

Example 5-29 Injection-Molded Part Suppose that the X and Y dimensions of an injection-molded part have
a bivariate normal distribution with σ X = 0.04, σ Y = 0.08, μ X = 3.00, μ Y = 7.70, and ρ = 0.8. Then
the probability that a part satisies both specii cations is

. (
.
.
.
P 2 95 < X < 3 05 7 60 <Y < 7 80)
,
This probability can be obtained by integrating f XY ( x,y;σ σ μ μ Y , )ρ over the region 2.95 < x < 3.05 and
,
,
,
X
Y
X
7.60 < y < 7.80, as shown in Fig. 5-3. Unfortunately, there is often no closed-form solution to probabilities involving
bivariate normal distributions. In this case, the integration must be done numerically.
EXERCISES FOR SECTION 5-3
Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.
(
(
5-48. Test results from an electronic circuit board indicate that (g) P W = ,X = 5) (h) P W = | X = 5)
5
5
50% of board failures are caused by assembly defects, 30% by
5-51. Four electronic ovens that were dropped during shipment
electrical components, and 20% by mechanical defects. Sup-

are inspected and classiied as containing either a major, a minor,
pose that 10 boards fail independently. Let the random vari-
or no defect. In the past, 60% of dropped ovens had a major
ables X, Y , and Z denote the number of assembly, electrical,
defect, 30% had a minor defect, and 10% had no defect. Assume
and mechanical defects among the 10 boards.
that the defects on the four ovens occur independently.
Calculate the following:
(a) P X = ,Y = ,Z = 2) (b) P X = 8) (a) Is the probability distribution of the count of ovens in
(
(
3
5
( |Y =1) ( |Y =1) each category multinomial? Why or why not?
(c) P X =8 (d) P X ≥ 8 (b) What is the probability that, of the four dropped ovens,
(
(e) P X = ,Y = | Z =7 1 2) two have a major defect and two have a minor defect?
5-49. Based on the number of voids, a ferrite slab is clas- (c) What is the probability that no oven has a defect?
siied as either high, medium, or low. Historically, 5% of the Determine the following:

slabs are classiied as high, 85% as medium, and 10% as low. (d) Joint probability mass function of the number of ovens
A sample of 20 slabs is selected for testing. Let X, Y , and Z with a major defect and the number with a minor defect
denote the number of slabs that are independently classii ed as (e) Expected number of ovens with a major defect
high, medium, and low, respectively. (f) Expected number of ovens with a minor defect
(a) What are the name and the values of the parameters of the (g) Conditional probability that two ovens have major defects
joint probability distribution of X, Y , and Z? given that two ovens have minor defects
(b) What is the range of the joint probability distribution of X, (h) Conditional probability that three ovens have major defects
Y , and Z? given that two ovens have minor defects
(c) What are the name and the values of the parameters of the (i) Conditional probability distribution of the number of ovens
marginal probability distribution of X? with major defects given that two ovens have minor defects
(d) Determine E X ( ) and V X ( ). (j) Conditional mean of the number of ovens with major defects
Determine the following: given that two ovens have minor defects.
(
(
(e) P X = 1 ,Y = 17 ,Z = ) 3 (f) P X ≤1 ,Y = 17 ,Z = ) 3 5-52. Let X and Y represent the concentration and viscosity of
a chemical product. Suppose that X and Y have a bivariate nor-
(
(g) P X ≤ ) 1 (h) E Y ( ) mal distribution with σ X = 4, σ Y = 1, μ X = 2 and μ y = 1. Draw a
(
(
(i) P X = 2 ,Z = Y | 3 = ) (j) P X = | Y = ) rough contour plot of the joint probability density function for
17
2
17
| (
(k) E X Y = ) each of the following values of r:
17
=
.
.
5-50. A Web site uses ads to route visitors to one of four landing (a) ρ 0 (b) ρ = 0 8 (c) ρ = −0 8
pages. The probabilities for each landing page are equal. Consider 5-53. Suppose that X and Y have a bivariate normal distribu-
3
7
.
8
.
20 independent visitors and let the random variables W, X, Y , and tion with σ X = 0 0. ,4 σ Y = 0 0 , μ X = 00, μ Y = .7 0, and ρ = 0.
Z denote the number of visitors routed to each page. Determine the following: . ( <Y < 7 80)
< X <3 05. )
(a) P 2 95. (
(b) P 7 60
Calculate the following: . ( <Y < 7 80) .
.
.
(
,
(a) P W = 5, X = 5, Y = 5, Z = ) (c) P 2 95 < X <3 05 7 60 .
5
(
(b) P W = 5, X = 5, Y = 5, Z = ) 5-54. In an acid-base titration, a base or acid is gradu-
5
ally added to the other until they have completely neutralized
(
(c) P W = 7, X = 7, Y = 6 Z = ) each other. Let X and Y denote the milliliters of acid and base
3
(
7
(d) P W = , X = ,Y =7 3 Z =3) needed for equivalence, respectively. Assume that X and Y have
σ = 2 mL,
a bivariate normal distribution with σ = 5 mL,
(e) P W( ≤ 2) (f) E W ( ) μ = 120 mL, μ = 100 mL, and ρ = .6. X Y
0
Y
X