CHAPTER 7B: RANDOM VARIABLES AND DISCRETE PROBABILITY
DISTRIBUTIONS
TRUE/FALSE
1. The Poisson probability distribution is a continuous probability distribution.
2. In a Poisson distribution, the mean and variance are equal.
3. The Poisson random variable is a discrete random variable with infinitely many possible values.
4. The mean of a Poisson distribution, where
is the average number of successes occurring in a
specified interval, is
.
5. The number of accidents that occur at a busy intersection in one month is an example of a Poisson
random variable.
6. The number of customers arriving at a department store in a 5-minute period has a Poisson
distribution.
7. The number of customers making a purchase out of 30 randomly selected customers has a Poisson
distribution.
8. The largest value that a Poisson random variable X can have is n.
9. The Poisson distribution is applied to events for which the probability of occurrence over a given span
of time, space, or distance is very small.
10. In a Poisson distribution, the variance and standard deviation are equal.
11. In a Poisson distribution, the mean and standard deviation are equal.
MULTIPLE CHOICE
1. Which of the following cannot have a Poisson distribution?
a.
The length of a movie.
b.
The number of telephone calls received by a switchboard in a specified time period.
c.
The number of customers arriving at a gas station in Christmas day.
d.
The number of bacteria found in a cubic yard of soil.
2. The Sutton police department must write, on average, 6 tickets a day to keep department revenues at
budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a
mean of 6.5 tickets per day. Interpret the value of the mean.
a.
The mean has no interpretation.
b.
The expected number of tickets written would be 6.5 per day.
c.
Half of the days have less than 6.5 tickets written and half of the days have more than 6.5
tickets written.
d.
The number of tickets that is written most often is 6.5 tickets per day.
3. The Poisson random variable is a:
a.
discrete random variable with infinitely many possible values.
b.
discrete random variable with finite number of possible values.
c.
continuous random variable with infinitely many possible values.
d.
continuous random variable with finite number of possible values.
4. Given a Poisson random variable X, where the average number of successes occurring in a specified
interval is 1.8, then P(X = 0) is:
a.
1.8
b.
1.3416
c.
0.1653
d.
6.05
5. In a Poisson distribution, the:
a.
mean equals the standard deviation.
b.
median equals the standard deviation.
c.
mean equals the variance.
d.
None of these choices.
6. On the average, 1.6 customers per minute arrive at any one of the checkout counters of Sunshine food
market. What type of probability distribution can be used to find out the probability that there will be
no customers arriving at a checkout counter in 10 minutes?
a.
Poisson distribution
b.
Normal distribution
c.
Binomial distribution
d.
None of these choices.
7. A community college has 150 word processors. The probability that any one of them will require
repair on a given day is 0.025. To find the probability that exactly 25 of the word processors will
require repair, one will use what type of probability distribution?
a.
Normal distribution
b.
Poisson distribution
c.
Binomial distribution
d.
None of these choices.
COMPLETION
1. In a Poisson experiment, the number of successes that occur in any interval of time is
____________________ of the number of success that occur in any other interval.
2. In a(n) ____________________ experiment, the probability of a success in an interval is the same for
all equal-sized intervals.
3. In a Poisson experiment, the probability of a success in an interval is ____________________ to the
size of the interval.
4. In Poisson experiment, the probability of more than one success in an interval approaches
____________________ as the interval becomes smaller.
5. A Poisson random variable is the number of successes that occur in a period of
____________________ or an interval of ____________________ in a Poisson experiment.
6. The ____________________ of a Poisson distribution is the rate at which successes occur for a given
period of time or interval of space.
7. In the Poisson distribution, the mean is equal to the ____________________.
8. In the Poisson distribution, the ____________________ is equal to the variance.
9. The possible values of a Poisson random variable start at ____________________.
10. A Poisson random variable is a(n) ____________________ random variable.
SHORT ANSWER
1. Compute the following Poisson probabilities (to 4 decimal places) using the Poisson formula:
a.
P(X = 3), if
= 2.5
b.
P(X 1), if
= 2.0
c.
P(X 2), if
= 3.0
2. Let X be a Poisson random variable with
= 6. Use the table of Poisson probabilities to calculate:
a.
P(X 8)
b.
P(X = 8)
c.
P(X 5)
d.
P(6 X 10)
b.
0.1032
0.7149
3. Let X be a Poisson random variable with
= 8. Use the table of Poisson probabilities to calculate:
a.
P(X 6)
b.
P(X = 4)
c.
P(X 3)
d.
P(9 X 14)
b.
0.0573
0.9862
911 Phone Calls
911 phone calls arrive at the rate of 30 per hour at the local call center.
4. {911 Phone Calls Narrative} Find the probability of receiving two calls in a five-minute interval of
time.
5. {911 Phone Calls Narrative} Find the probability of receiving exactly eight calls in 15 minutes.
ANS:
6. {911 Phone Calls Narrative} If no calls are currently being processed, what is the probability that the
desk employee can take four minutes break without being interrupted?
Classified Department Phone Calls
A classified department receives an average of 10 telephone calls each afternoon between 2 and 4 P.M.
The calls occur randomly and independently of one another.
7. {Classified Department Phone Calls Narrative} Find the probability that the department will receive
13 calls between 2 and 4 P.M. on a particular afternoon.
8. {Classified Department Phone Calls Narrative} Find the probability that the department will receive
seven calls between 2 and 3 P.M. on a particular afternoon.
9. {Classified Department Phone Calls Narrative} Find the probability that the department will receive at
least five calls between 2 and 4 P.M. on a particular afternoon.
Post office
10. {Post Office Narrative} Find the probability that the number of arrivals between 3:00 and 5:00 P.M. is
at least 10.
11. {Post Office Narrative} Find the probability that the number of arrivals between 3:30 and 4:00 P.M. is
at least 10.
12. {{Post Office Narrative} Find the probability that the number of arrivals between 4:00 and 5:00 P.M.
is exactly two.
13. Suppose that the number of buses arriving at a Depot per minute is a Poisson process. If the average
number of buses arriving per minute is 3, what is the probability that exactly 6 buses arrive in the next
minute?
Unsafe Levels of Radioactivity
The number of incidents at a nuclear power plant has a Poisson distribution with a mean of 6 incidents
per year.
14. {Unsafe Levels of Radioactivity Narrative} Find the probability that there will be exactly 3 incidents
in a year.
15. {Unsafe Levels of Radioactivity Narrative} Find the probability that there will be at least 3 incidents
in a year.
16. {Unsafe Levels of Radioactiviy Narrative} Find the probability that there will be at least 1 incident in
a year.
17. {Unsafe Levels of Radioactivity Narrative} Find the probability that there will be no more than 1
incident in a year.
18. {Unsafe Levels of Radioactivity Narrative} Find the variance of the number of incidents in one year.
19. {Unsafe Levels of Radioactivity Narrative} Find the standard deviation of the number of incidents is
in one year.
