Question 1

The local police department must write, on average, 5 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 number of tickets that is written most often is 6.5 tickets per day.
B) Half of the days have less than 6.5 tickets written and half of the days have more than 6.5 tickets written.
C) If we sampled all days, the arithmetic average or expected number of tickets written would be 6.5 tickets per day.
D) The mean has no interpretation since 0.5 ticket can never be written.

Accepted Answer

The value of the mean (6.5 tickets per day) represents the expected number of tickets written per day if we were to take a large sample of days. In a Poisson distribution, the mean is the same as the variance, so we can expect that on some days there may be more than 6.5 tickets written and on other days there may be less, but on average we would expect 6.5 tickets to be written per day.

Question 2

TABLE 5-2
A certain type of new business succeeds 60% of the time. Suppose that three such businesses open (where they do not compete with each other, so it is reasonable to believe that their relative successes would be independent).
-Referring to Table 5-2, the probability that at least one business succeeds is ________.

Accepted Answer

The answer of TABLE 5-2
A certain type of new business...

Question 3

The diameters of 10 randomly selected bolts have a binomial distribution.

Accepted Answer

The diameters of bolts are not binary outcomes. Therefore, a binomial distribution is not appropriate for modeling their diameters.

Question 4

TABLE 5-2
A certain type of new business succeeds 60% of the time. Suppose that three such businesses open (where they do not compete with each other, so it is reasonable to believe that their relative successes would be independent).
-Referring to Table 5-2, the probability that all three businesses succeed is ________.

Accepted Answer

The answer of TABLE 5-2
A certain type of new business...

Question 5

TABLE 5-2
A certain type of new business succeeds 60% of the time. Suppose that three such businesses open (where they do not compete with each other, so it is reasonable to believe that their relative successes would be independent).
-Referring to Table 5-2, the probability that all three businesses fail is ________.

Accepted Answer

The answer of TABLE 5-2
A certain type of new business...

Question 6

Suppose that a judge's decisions follow a binomial distribution and that his verdict is incorrect 10% of the time. In his next 10 decisions, the probability that he makes fewer than 2 incorrect verdicts is 0.736.

Accepted Answer

Let X be the number of incorrect verdicts the judge makes in the next 10 decisions.
X ~ Bin(10, 0.1)
We want to find P(X < 2) which is equal to P(X = 0) + P(X = 1)

Using the binomial formula:
P(X = 0) = (10 choose 0) * 0.1^0 * 0.9^10 = 0.3487
P(X = 1) = (10 choose 1) * 0.1^1 * 0.9^9 = 0.3874

Therefore, P(X < 2) = 0.3487 + 0.3874 = 0.7361

Since the given probability matches with our calculation, the statement is true.

Question 7

Another name for the mean of a probability distribution is its expected value.

Accepted Answer

This statement is true. The expected value is the same as the mean of a probability distribution.

Question 8

The largest value that a Poisson random variable X can have is n.

Accepted Answer

A Poisson random variable can take on any non-negative integer value, so it does not have a maximum value of n.

Question 9

TABLE 5-1
The probability that a particular type of smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8. You have two such alarms in your home and they operate independently.
-Referring to Table 5-1, the probability that both sound an alarm in the presence of smoke is ________.

Accepted Answer

The answer of TABLE 5-1
The probability that a particular type...

Question 10

TABLE 5-1
The probability that a particular type of smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8. You have two such alarms in your home and they operate independently.
-Referring to Table 5-1, the probability that neither sound an alarm in the presence of smoke is ________.

Accepted Answer

The answer of TABLE 5-1
The probability that a particular type...

Question 11

In a Poisson distribution, the mean and standard deviation are equal.

Accepted Answer

The answer of In a Poisson distribution, the mean and...

Question 12

If π remains constant in a binomial distribution, an increase in n will increase the variance.

Accepted Answer

The variance of a binomial distribution is given by npq, where n is the number of trials, p is the probability of success, and q is the probability of failure. If π (or p) remains constant and n increases, then the product np increases, which in turn increases the variance.

Question 13

TABLE 5-2
A certain type of new business succeeds 60% of the time. Suppose that three such businesses open (where they do not compete with each other, so it is reasonable to believe that their relative successes would be independent).
-Referring to Table 5-2, the probability that exactly one business succeeds is ________.

Accepted Answer

The answer of TABLE 5-2
A certain type of new business...

Question 14

In a Poisson distribution, the mean and variance are equal.

Accepted Answer

This is true for all Poisson distributions. The mean and variance of a Poisson distribution are both equal to lambda, the parameter that represents the average number of occurrences in the given time interval or space.

Question 15

TABLE 5-1
The probability that a particular type of smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8. You have two such alarms in your home and they operate independently.
-Referring to Table 5-1, the probability that at least one sounds an alarm in the presence of smoke is ________.

Accepted Answer

The answer of TABLE 5-1
The probability that a particular type...

Question 16

The number of customers arriving at a department store in a five-minute period has a binomial distribution.

Accepted Answer

A binomial distribution is used for a fixed number of independent trials with two possible outcomes. The number of customers arriving is better modeled by a Poisson distribution, which is used for counting the number of events in a fixed interval of time or space.

Question 17

The Poisson distribution can be used to model a continuous random variable.

Accepted Answer

The Poisson distribution is used to model a discrete random variable, not a continuous random variable.

Question 18

Suppose that the number of airplanes arriving at an airport per minute is a Poisson process. The average number of airplanes arriving per minute is 3. The probability that exactly 6 planes arrive in the next minute is 0.05041.

Accepted Answer

The answer of Suppose that the number of airplanes arriving...

Question 19

The number of customers arriving at a department store in a five-minute period has a Poisson distribution.

Accepted Answer

The Poisson distribution is often used to model the number of arrivals of customers or events in a given time period. Therefore, it is true that the number of customers arriving at a department store in a five-minute period has a Poisson distribution.

Question 20

If π remains constant in a binomial distribution, an increase in n will not change the mean.

Accepted Answer

The answer of If π remains constant in a binomial...

Quiz 5: Discrete Probability Distributions

The local police department must write, on average, 5 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.

The diameters of 10 randomly selected bolts have a binomial distribution.

Suppose that a judge's decisions follow a binomial distribution and that his verdict is incorrect 10% of the time. In his next 10 decisions, the probability that he makes fewer than 2 incorrect verdicts is 0.736.

Another name for the mean of a probability distribution is its expected value.

The largest value that a Poisson random variable X can have is n.

In a Poisson distribution, the mean and standard deviation are equal.

If π remains constant in a binomial distribution, an increase in n will increase the variance.

In a Poisson distribution, the mean and variance are equal.

The number of customers arriving at a department store in a five-minute period has a binomial distribution.

The Poisson distribution can be used to model a continuous random variable.

Suppose that the number of airplanes arriving at an airport per minute is a Poisson process. The average number of airplanes arriving per minute is 3. The probability that exactly 6 planes arrive in the next minute is 0.05041.

The number of customers arriving at a department store in a five-minute period has a Poisson distribution.

If π remains constant in a binomial distribution, an increase in n will not change the mean.