Determine whether each of the following is a valid probability distribution

ECN 422 PS2
7.5 Determine whether each of the following is a valid probability distribution.
a)
x 0 1 2 3
P(x) 0.1 0.3 0.4 0.1
b)
x 5 −6 10 0
P(x) 0.01 0.01 0.01 0.97
c)
x 14 12 −7 13
P(x) 0.25 0.46 0.04 0.24
7.7 In a recent census, the number of color televisions per household was recorded
Number of color televisions 0 1 2 3 4 5
Number of households (thousands) 1218 32379 37961 19387 7714 2842
a) Develop the probability distribution of X, the number of color televisions per household. For
part (a), draw both the p.m.f. table and the p.m.f. graph.
b) Determine the following probabilities.
P(X ≤ 2), P(X > 2), P(X ≥ 4)
7.9 Second-year business students at many universities are required to take 10 one-semester courses.
The number of courses that result in a grade of A is a discrete random variable. Suppose that each
value of this random variable has the same probability. Determine the probability distribution.
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ECN 422 PS2 Zhanhan Yu
7.11 An Internet pharmacy advertises that it will deliver the over-the-counter products that customers purchase in 3 to 6 days. The manager of the company wanted to be more precise in its
advertising. Accordingly, she recorded the number of days it took to deliver to customers. From
the data, the following probability distribution was developed.
Number of days 0 1 2 3 4 5 6 7 8
Probability 0 0 0.01 0.04 0.28 0.42 0.21 0.02 0.02
a) What is the probability that a delivery will be made within the advertised 3- to 6-day period?
b) What is the probability that a delivery will be late?
c) What is the probability that a delivery will be early?
7.13 The probability that a university graduate will be offered no jobs within a month of graduation
is estimated to be 5%. The probability of receiving one, two, and three job offers has similarly been
estimated to be 43%, 31%, and 21%, respectively. Determine the following probabilities.
a) A graduate is offered fewer than two jobs.
b) A graduate is offered more than one job.
c) Also draw both the p.m.f. table and the p.m.f. graph.
7.19 We are given the following probability distribution.
x 0 1 2 3
P(x) 0.4 0.3 0.2 0.1
a) Calculate the mean, variance, and standard deviation.
b) Suppose that Y = 3X + 2 For each value of X, determine the value of Y. What is the probability
distribution of Y?
c) Calculate the mean, variance, and standard deviation from the probability distribution of Y.
d) Use the laws of expected value and variance to calculate the mean, variance, and standard
deviation of Y from the mean, variance, and standard deviation of X. Compare your answers in
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parts (c) and (d). Are they the same (except for rounding)?
7.47 The bivariate distribution of X and Y is described here.
x
y 1 2
1 0.28 0.42
2 0.12 0.18
a) Find the marginal probability distribution of X.
b) Find the marginal probability distribution of Y.
c) Compute the mean and variance of X.
d) Compute the mean and variance of Y.
7.48 Refer to Exercise 7.47. Compute the covariance and the coefficient of correlation.
7.49 Refer to Exercise 7.47. Use the laws of expected value and variance of the sum of two
variables to compute the mean and variance of X + Y .
7.50 Refer to Exercise 7.47.
a) Determine the distribution of X + Y .
b) Determine the mean and variance of X + Y .
c) Does your answer to part (b) equal the answer to Exercise 7.49?
7.54 After analyzing several months of sales data, the owner of an appliance store produced the
following joint probability distribution of the number of refrigerators and stoves sold daily.
Refrigerators
Stoves 0 1 2
0 0.08 0.14 0.12
1 0.09 0.17 0.13
2 0.05 0.18 0.04
a) Find the marginal probability distribution of the number of refrigerators sold daily.
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ECN 422 PS2 Zhanhan Yu
b) Find the marginal probability distribution of the number of stoves sold daily. c) Compute the
mean and variance of the number of refrigerators sold daily.
d) Compute the mean and variance of the number of stoves sold daily.
e) Compute the covariance and the coefficient of correlation.
7.56 Refer to Exercise 7.54. Find the following conditional probabilities.
a) P(1refrigerator|0stoves)
b) P(0stoves|1refrigerator)
c) P(2refrigerators|2stoves)
7.87 Given a binomial random variable with n = 6 and p = .2 use the formula to find the following
probabilities.
a) P(X = 2)
b) P(X = 3)
c) P(X = 5)
7.88 Repeat Exercise 7.87 using Table 1 in Appendix B (Binomial Probabilities Table).
7.94 A certain type of tomato seed germinates 90% of the time. A backyard farmer planted 25
seeds. a) What is the probability that exactly 20 germinate?
b) What is the probability that 20 or more germinate?
c) What is the probability that 24 or fewer germinate?
d) What is the expected number of seeds that germinate?
7.96 A student majoring in accounting is trying to decide on the number of firms to which he
should apply. Given his work experience and grades, he can expect to receive a job offer from
70% of the firms to which he applies. The student decides to apply to only four firms. What is the
probability that he receives no job offers?
7.97 In the United States, voters who are neither Democrat nor Republican are called Independents. It is believed that 10% of all voters are Independents. A survey asked 25 people to identify
themselves as Democrat, Republican, or Independent.
a) What is the probability that none of the people are Independent?
b) What is the probability that fewer than five people are Independent?
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ECN 422 PS2 Zhanhan Yu
c) What is the probability that more than two people are Independent?
7.127 The percentage of customers who enter a restaurant and ask to be seated in a smoking section
is 15%. Suppose that 100 people enter the restaurant.
a) What is the expected number of people who request a smoking table?
b) What is the standard deviation of the number of requests for a smoking table?
c) What is the probability that 20 or more people request a smoking table?
Read Keller Exercise 7.127.
(a) Let X be the number of people who request a smoking table. What is the distribution of X?
You need to specify all parameter values of the distribution.
(b) Answer to a), b) and c).
8.4 A random variable is uniformly distributed between 5 and 25.
a) Draw the density function.
b) Find P(X > 25)
c) Find P(10 < X < 15)
d) Find P(5.0 < X < 5.1)
8.6 The amount of time it takes for a student to complete a statistics quiz is uniformly distributed
between 30 and 60 minutes. One student is selected at random. Find the probability of the following events.
a) The student requires more than 55 minutes to complete the quiz.
b) The student completes the quiz in a time between 30 and 40 minutes.
c) The student completes the quiz in exactly 37.23 minutes.
8.34 X is normally distributed with mean 100 and standard deviation 20. What is the probability
that X is greater than 145?
8.35 X is normally distributed with mean 250 and standard deviation 40. What value of X does
only the top 15% exceed?
8.36 X is normally distributed with mean 1,000 and standard deviation 250. What is the probability
that X lies between 800 and 1,100?
8.46 The heights of 2-year-old children are normally distributed with a mean of 32 inches and a
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ECN 422 PS2 Zhanhan Yu
standard deviation of 1.5 inches. Pediatricians regularly measure the heights of toddlers to determine whether there is a problem. There may be a problem when a child is in the top or bottom 5%
of heights. Deter-mine the heights of 2-year-old children that could be a problem.
8.47 Refer to Exercise 8.46. Find the probability of these events.
a) A 2-year-old child is taller than 36 inches.
b) A 2-year-old child is shorter than 34 inches.
c) A 2-year-old child is between 30 and 33 in ches tall.
8.50 The amount of time devoted to studying statistics each week by students who achieve a grade
of A in the course is a normally distributed random variable with a mean of 7.5 hours and a standard deviation of 2.1 hours.
a) What proportion of A students study for more than 10 hours per week?
b) Find the probability that an A student spends between 7 and 9 hours studying.
c) What proportion of A students spend fewer than 3 hours studying?
d) What is the amount of time below which only 5% of all A students spend studying?
8.51 The number of pages printed before replacing the cartridge in a laser printer is normally distributed with a mean of 11,500 pages and a standard deviation of 800 pages. A new cartridge has
just been installed.
a) What is the probability that the printer produces more than 12,000 pages before this cartridge
must be replaced?
b) What is the probability that the printer produces fewer than 10,000 pages?
8.52 Refer to Exercise 8.51. The manufacturer wants to provide guidelines to potential customers
advising them of the minimum number of pages they can expect from each cartridge. How many
pages should it advertise if the company wants to be correct 99% of the time?
8.54 Because of the relatively high interest rates, most consumers attempt to pay off their credit
card bills promptly. However, this is not always possible. An analysis of the amount of interest
paid monthly by a bank’s Visa cardholders reveals that the amount is normally distributed with a
mean of $27 and a standard deviation of $7.
a) What proportion of the bank’s Visa cardholders pay more than $30 in interest?
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b) What proportion of the bank’s Visa cardholders pay more than $40 in interest?
c) What proportion of the bank’s Visa cardholders pay less than $15 in interest?
d) What interest payment is exceeded by only 20% of the bank’s Visa cardholders?
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