Discrete Probability Distributions
1.
Suppose a statistician working for CSULA Federal Credit Union collected
data on ATM withdrawals for the population of the credit union’s
customers. The statistician generates the following probability distribution
of withdrawals for a specific day.
Number of
Probability
Withdrawals
0
0.53
1
0.26
2
0.16
3
0.04
4
0.01
A.
B.
C.
D.
E.
F.
G.
Define what the population is.
The population is all customers at the CSULA Federal Credit Union.
The characteristic we are concerned with is the customers’ withdrawals
on a specific day.
Define what the population mean is for the problem.
The population mean is the average number of withdrawals on a
particular for the population.
Calculate the expected number of ATM withdrawals made on the
specific day.
The expected number of ATM withdrawals: μ=.74
What is the median number of withdrawals made that day?
Zero.
Calculate the variance and standard deviation in the number of
withdrawals that day.
The population variance is .8724. The standard deviation is .934.
What is P(X>1)?
.21
What is P(X≤2)?
.95
2.
A new math book costs $140 and a used one costs $80. A new chemistry
book costs $180, a good used one costs $75, and a worn one costs $40. A
student wants to buy the cheapest math book and chemistry book available
in the bookstore. The probability of getting a used math book is .4, of
getting a worn chemistry book is .3, and of getting a good used chemistry
book is .2. Let X denote the cost of the two books the student purchases.
Assume that the purchases are independent of one another. Construct the
probability distribution of X.
X
P(X)
120
0.12
155
0.08
180
0.18
215
0.12
260
0.20
320
0.30
3.
Suppose you agree to a bet involving a single toss of a fair coin. If the coin
winds up heads, you win $200; if it is tails, you lose $100. Define X as the
outcome of the bet for you. Construct the probability distribution of X.
What is the expected value of X? What is the probability you will lose
money?
Event
X
P(X)
T
-100
0.50
H
+200
0.50
What is the expected value of X? E(X)=$50.
What is the probability you will lose money? 0.50
Now suppose the bet involves two coin tosses. The previously stated
outcomes apply for each toss (win $200 if heads, lose $100 if tails). Define
X as the dollar value of the intersection of events for the coin tosses.
Construct the probability distribution of X.
Event
TT
TH
HT
HH
X
-200
+100
+100
+400
P(X)
.25
.25
.25
.25
What is the expected value of X? E(X)=$100.
What is the probability you will lose money? 0.25
4.
Farming is a very risky business because of the possibility that bad weather
might cause financial ruin. Farmers usually insure their crops each year
against possible losses resulting from bad weather. Determine the annual
premium for an insurance policy to cover the farmer's $200,000 wheat crop.
Insurance representatives believe there is a 3% probability that the crop will
be totally destroyed by adverse weather, and a 5% probability of a
$100,000 loss due to weather.
Let X represent the expected loss to the farmer.
X
0
$100,000
$200,000
P(X)
0.92
0.05
0.03
The expected value of X is $11,000. This equals the expected cost to the
insurer, in which case the insurance premium will be at least $11,000.
5.
Suppose you have been offered one share of stock from your boss as a
Christmas bonus. She offers you a choice of Wal Mart or Sears stock. You
know that you are going to hold the stock for one year. Below are the
probability distributions of the two stocks’ prices a year from today.
Wal
Mart
Stock
$40
$50
$100
A.
B.
C.
Probability Sears Probability
Stock
0.25
.50
0.25
$45
$50
$120
0.10
0.80
.10
Fill in the above table.
What is the expected price for the two stocks?
The expected value of the Wal Mart stock is $60 after a year. The expected
value of the Sears stock is $57.
What is the variance and standard deviation for the two stocks' prices?
Wal Mart
Sears
Stock
Stock
variance
550
450.25
standard
23.45
21.22
deviation
D.
Why should variance/standard deviation be considered in determining
which stock to acquire?
The variance and standard deviation are simple measures of the risk of
holding a financial asset.