1
ISyE 3770 – Fall 2007
Homework #5 (Covers Modules 16–18) — Solutions
1. (Hines et al., 4–1). A refrigerator manufacturer subjects his finished products to
a final inspection. Of interest are two categories of defects: scratches or flaws in
the porcelain finish, and mechanical defects. The number of each type of defects
is a random variable. The results of inspecting 50 refrigerators are shown in the
following joint p.m.f. table, where
X
represents the occurrence of finish defects and
Y
represents the occurrence of mechanical defects.
Y
\
X
0
1
2
3
4
5
0
11/50
4/50
2/50
1/50
1/50
1/50
1
8/50
3/50
2/50
1/50
1/50
2
4/50
3/50
2/50
1/50
3
3/50
1/50
4
1/50
(a) Find the marginal distributions of
X
and
Y
.
Solution:
Let’s rewrite the table, this time including the marginals.
Y
\
X
0
1
2
3
4
5
f
Y
(
y
)
0
11/50
4/50
2/50
1/50
1/50
1/50
20/50
1
8/50
3/50
2/50
1/50
1/50
15/50
2
4/50
3/50
2/50
1/50
10/50
3
3/50
1/50
4/50
4
1/50
1/50
f
X
(
x
)
27/50
11/50
6/50
3/50
2/50
1/50
♦
(b) Find the marginal distribution of mechanical defects, given that there are no
finish defects.
Solution:
f
(
y

X
= 0) =
f
(0
, y
)
f
X
(0)
=
f
(0
, y
)
27
/
50
=
11
/
27
if
y
= 0
8
/
27
if
y
= 1
4
/
27
if
y
= 2
3
/
27
if
y
= 3
1
/
27
if
y
= 4
0
otherwise
♦
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
(c) Find the marginal distribution of finish defects, given that there are no me
chanical defects.
Solution:
f
(
x

Y
= 0) =
f
(
x,
0)
f
Y
(0)
=
f
(
x,
0)
20
/
50
=
11
/
20
if
x
= 0
4
/
20
if
x
= 1
2
/
20
if
x
= 2
1
/
20
if
x
= 3
,
4
,
5
0
otherwise
♦
2. (Hines et al., 4–4). Consider a situation in which the surface tension and acidity
of a chemical product are measured. These variables are coded such that surface
tension is measured on a scale 0
≤
X
≤
2, and acidity is measured on a scale
2
≤
Y
≤
4. The probability density function of (
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 goldsman
 probability density function, dx dy, Hines

Click to edit the document details