Mathematics for Business Decisions, Part II
Homework 6
for
Math 115b, Section:
Instructor: James Barrett
Date:
by
Team:
We, the undersigned, maintain that each of us participated fully and equally in the completion of
this assignment and that the work contained herein is original. Furthermore, we acknowledge
that sanctions will be imposed jointly if any part of this work is found to violate the Student
Code of Conduct, the Code of Academic Integrity, or the policies and procedures established for
this course.
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1. Let X be the random variable whose c.d.f. is given below.
0.0
0.2

FX ( x)  0.5
0.7

1.0
if
if
if
if
if
x2
2 x4
4 x6
6 x8
8 x
(i) Find the p.m.f for X.
(ii) Find the following probabilities:
P2  X  6
P( X  8)
P ( X  4)
P( X  5)
P ( X  7)
P( X  2)
(iii) Compute the E(X).
Solution:
2. Let X be a binomial random variable with n  20 and p  0.65 . Use the
Binomial Distribution function in Excel to compute (i) f X (14) and (ii) FX (14) . (iii)
Plot a graph of the p.m.f., f X (x) . (iv) Plot a graph of the c.d.f., FX (x) . Your plots
should resemble those made in Graphs of Binomial 2.xls.
Solution:
3. Consolidated Widgets Inc. has a force of 130 salespeople, whose sales
performances can be considered to be independent of each other. The manager
estimates that the probability of any one sales person reaching his or her monthly
quota is 0.6. (i) What is the probability that at least 70 people make their quota?
(ii) What is the probability that at most 70 people make their quota?
Solution:
4. Use Experimentation in Binomial 2.xlsx and the situation in Example 2 to find
the smallest value of p, rounded to 2 decimal places, that the saleswoman would
need in order for the probability of at least 7 sales per day to be greater than or
equal to 0.6. Assume that she makes 32 contacts daily
Solution
5. Refer to Problem 3. Let Q be the number of salespeople who make their quotas
in a given month. (i) Use the special formula for the expected value of a binomial
random variable to compute E (Q ) . (ii) In the real world language of people and
sales, what is the interpretation of the expected value of Q?
Solution:
6. Use Integrating.xlsm to determine whether or not
 x 2  2  x if 0  x  2
f ( x)  
could be a p.d.f. for some continuous random
0
elsewhere
variable.
Solution:
7. Use the p.d.f. and c.d.f. defined in Example 5. Compute the probability that the
time between computer system shutdowns will be at least 24 hours. (i)
Do
this with the p.d.f, using Integrating.xlsm (ii) Do this with the c.d.f.,.
Solution:
8. Let X be an exponential random variable with parameter   0.5 . (i) Use FX to
compute P(0.2  X  1.2) . (ii) Use f X to compute the same probability.
Solution:
9. The lifetime, in years, of the transmitter in your new wireless mouse is an
exponential random variable with parameter   3.8 years. What is the probability
that the transmitter lasts for at least 4 and 1/2 years.
Solution:
10. Let X be a continuous random variable that is uniform on the interval [0, 10] .
(i) What is the probability the X is at most 6.87? (ii) What is the probability that X
is no less than 5.59?
Solution:
11. Let W be the working lifetime, measured in years, of the microchip in your
new digital watch. The manufacturer claims that W has an exponential
distribution with parameter   5 years. (i) Assuming that this is correct, use
Integrating.xlsm and the probability density function fW to verify that the
expected value of W is 5 years. Compute the probabilities that the chip lasts for (ii)
at least 9 years and (iii) at most 4 years. Do two way using f(x) and Integrating
.xlsm and by using F(x).
Solution:
12. Let W be the random variable described in Problem 11. Suppose that the
actual mean of W is 3-1/2 years. Use Integrating.xlsm and the probability density
function fW to compute the probabilities that the chip lasts for (i) at least 9 years
and (ii) at most 4 years. Also answer each question using F(x).
Solution:
13. Let B be the random variable giving the number of checks that your business
takes in per month that are returned for insufficient funds. Your records for the
last 8 months show 14, 19, 8, 8, 21, 12, 17, and 6 such checks. (i) Use this
information to approximate the expected value of B. (ii) What is the real world
interpretation of your answer to Part (i)? (iii) Is your answer to Part (i) the exact
value of E (B ) ? Explain your answer.
Solution.
14. Let S be the continuous random variable giving your company’s weekly gross
sales in thousands of dollars. Records contain the sample values of S that are given
in the sheet Exercise Data 2 of Samples.xlsx. (i) Compute the minimum,
maximum, and mean of the sample. (ii) Select suitable bin limits and use the
HISTOGRAM function to plot an approximation of the p.d.f., 𝑓𝑆 . (iii) Do you
think that it is very likely that S is an exponential random variable?
Solution:
15. Use Excel and your team’s data for the signals on past oil leases to compute
the complete set of errors in the signals. Find the sample mean of these errors and
plot a histogram which approximates the actual p.d.f., 𝑓𝑅 , of R.
Solution.