This preview shows pages 1–4. Sign up to view the full content.
Exemplars for the Homework Problems from Chapter 5
Required for Homework #6
Problems 14, 17 (with additional questions), and 21 (with 10)
14.
Let
X
be the discrete r.v. of profit, in $K, with values
x
= 100, 0, 50, 100,
150, and 200.
a.
To be a valid probability distribution,
prob
(100) +
prob
(0) +
prob
(50) +
prob
(100) +
prob
(150) +
prob
(200) = 1
So
Prob
(
x
= 200)
= 1 – [
prob
(
x
= 100) +
prob
(
x
= 0) +
prob
(
x
= 50) +
prob
(
x
= 100)
+
prob
(
x
= 150)]
= 1 – (0.10 + 0.20 + 0.30 + 0.25 + 0.10)
= 1 – 0.95
=
0.05
The probability that MRA will have a $200,000 profit is 0.05.
In other
words, there is a 5% chance that MRA will have a $200,000 profit.
The probability distribution for profit x is:
x
prob(x
)
100
0.10
0
0.20
50
0.30
100
0.25
150
0.10
200
0.05
1.00
b.
“Profit” means
x
> 0, so
Prob
(Profit) =
prob
(
x
> 0)
= prob
(
x
= 50) +
prob
(
x
= 100) +
prob
(
x
= 150)
HW #6 exemplars
1
© 2010 Harvey Singer
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document +
prob
(
x
= 200)
200)
=
0.30 + 0.25 +
0.10 + 0.05
=
0.70
The probability that MRA will have a profit is 0.70.
In other words, there is a
70% chance that MRA will be profitable.
c.
The phrase “at least $100,000” means “$100,000 or more.”
The discrete
values of
x
that are “100,000 or more” are
x
= 100, 150, and 200.
Prob
(at least 100) =
prob
(
x
≥ 100)
= prob
(
x
= 100) +
prob
(
x
= 150) +
prob
(
x
= 200)
=
0.25 + 0.10 + 0.05
=
0.40
The probability that MRA will have a profit of at least $100,000 is 0.40.
In
other words, there is a 40% chance that MRA will have a profit of at least
$100,000.
(Another way of stating this: There is a 40% chance that MRA will have a
profit of at not less than $100,000.)
*********************************************************
New questions:
1. How much profit should be expected?
Calculate expected value of
x
from the probability distribution:
x
prob(x
)
x*prob(x
)
100
0.10
10
0
0.20
0
50
0.30
15
100
0.25
25
150
0.10
15
200
0.05
10
1.00
55.00
HW #6 exemplars
2
© 2010 Harvey Singer
Expect a profit of $55,000.
Of course, the profit may be more or it may be
less, but as a first an best guess, expect a profit of $55K.
Notes:
a. The expected value is NOT the arithmetic average (mean) of the values of
the r.v.
x
.
That is, the expected value of
x
is NOT the 66.67, the arithmetic
average of the values
x
= 100, 0, 50, 100, 150, and 200.
This is because the values are not all equally likely, that is, they do not have
the same chances of occurrence.
For randomly sampled data, it is assumed
that all the values are equally likely, as was done in Chapter 3 (although not
stated at the time).
But not here.
These are not randomly sampled values.
They are all possible
values of the random variable
X
.
Some of the values of
x
are more likely than
others.
As a result, the probability of a value, and not just the value itself,
also drives what should be expected.
b. The r.v.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 02/19/2011 for the course OM 210 taught by Professor Singer during the Spring '08 term at George Mason.
 Spring '08
 SINGER

Click to edit the document details