Page 192 - Applied statistics and probability for engineers

170 Chapter 5/Joint Probability Distributions

Example 5-18 Layer Thickness Suppose that X , X ,and X 3 represent the thickness in micrometers of a
2
1
substrate, an active layer, and a coating layer of a chemical product, respectively. Assume that

X , X ,and X 3 are independent and normally distributed with μ = 10000, μ = 1000, μ = 80, σ = 250, σ = 20,
2
2
1
1
1
2
3
and σ = 4, respectively. The specii cations for the thickness of the substrate, active layer, and coating layer are
3
,
9200 < x < 10 800 950 < x < 1050, and 75< x < 85, respectively. What proportion of chemical products meets all
,
1
2
3

thickness speciications? Which one of the three thicknesses has the least probability of meeting specii cations?
(
The requested probability is P 9200 < X <10 ,800 , 950 < X < 1050 75 < X < 85) . Because the random variables
,
1
2
3
are independent,
(
,
,
P 9200
P 9200 < X <10 ,800 950 < X <1050 75 < X <85) = ( < X <10 ,800)
1
1
2
3
× × P(950 < X <1050 ) P (75 < X <85 )
3
2
After standardizing, the above equals
(
1 25
P − .5
3 2
2 5
P − .2 < Z < . ) ( 2 < Z < . ) ( 1 < Z < . )
P − .25
3
where Z is a standard normal random variable. From the table of the standard normal distribution, the requested
probability equals
)(
)(
. (
.
.
0 99862 0 98758 0 78870) = 0 7778
.
The thickness of the coating layer has the least probability of meeting speciications. Consequently, a priority should

be to reduce variability in this part of the process.
EXERCISES FOR SECTION 5-1
Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.
(
(
(

5-1. Show that the following function satisies the proper- (e) E X , ) E Y , ) V X , ) and V Y ( )
ties of a joint probability mass function. (f) Marginal probability distribution of X
(g) Conditional probability distribution of Y given that X = 1
x y f XY ( x, y) (h) Conditional probability distribution of X given that Y = 2
| (
1.0 1 1 4 (i) E Y X = ) 1 (j) Are X and Y independent?

1.5 2 1 8 5-3. Show that the following function satisies the proper-
ties of a joint probability mass function.
1.5 3 1 4
x y f XY ( x, y)
2.5 4 1 4
–1.0 –2 1 8
3.0 5 1 8
–0.5 –1 1 4
Determine the following: 0.5 1
(
(
.
(a) P X < 2 5 . ,Y <3) (b) P X < 2 5) 1 2
(
(
(c) P Y <3) (d) P X >1 8 ,Y > 4 7) 1.0 2 1 8
.
.
(
( ) (
,Y <1 5)
(
(

(e) E X ( ), E Y ,V X , ) and V Y). Determine the following: (b) P X < 0 5)
.
.
(a) P X < 0 5 .
(
(f) Marginal probability distribution of X (c) P Y <1 5) (d) P X > 0 25 ,Y < 4 5)
(
.
.
.
(g) Conditional probability distribution of Y given that X = 1.5 (e) E X , ) E Y , ) V X , ) and V Y ( )
(
(
(
(h) Conditional probability distribution of X given that Y = 2
| (
1
(i) E Y X = . ) 5 (j) Are X and Y independent? (f) Marginal probability distribution of X
(g) Conditional probability distribution of Y given that X = 1
5-2. Determine the value of c that makes the function
f x, y ( ) = ( y) a joint probability mass function over the (h) Conditional probability distribution of X given that Y = 1
c x +
| (
nine points with x = 1 2 3, , and y = 1 2 3, , . (i) E X y = ) 1 (j) Are X and Y independent?
Four electronic printers are selected from a large lot
5-4.
Determine the following: of damaged printers. Each printer is inspected and classii ed
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(
(a) P X = 1 ,Y < ) 4 (b) P X = ) 1 as containing either a major or a minor defect. Let the random
(
(

(c) P Y = ) 2 (d) P X < ,Y < 2) variables X and Y denote the number of printers with major
2