Question 1

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

Accepted Answer

The Poisson distribution is used to model discrete random variables, not continuous ones.

Question 2

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, including infinite possibilities. Therefore, there is no maximum value n that a Poisson random variable can have.

Question 3

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

Accepted Answer

This is true. In a Poisson distribution, the mean (λ) is also the variance (λ).

Question 4

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

Accepted Answer

The diameters of the bolts would not have a binomial distribution as it is a continuous variable rather than a discrete variable. A binomial distribution is determined by the number of successes in a fixed number of trials, where each trial has only two possible outcomes (success or failure). The diameter of a bolt can have an infinite number of possible outcomes, making it a continuous variable.

Question 5

SCENARIO 5-1
The probability that a smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8.You have 2 such alarms in your home and they operate independently.
-Referring to Scenario 5-1, the probability that neither sound an alarm in the presence of smoke is_.

Accepted Answer

The answer of SCENARIO 5-1
The probability that a smoke alarm...

Question 6

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

Accepted Answer

In a binomial distribution, the mean is given by μ = n*p, where n is the number of trials and p is the probability of success. Therefore, if n increases, the mean will also increase, assuming p remains constant.

Question 7

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 in his next 10 decisions. Then X follows a binomial distribution with n = 10 and p = 0.1. 
We are asked to find P(X<2), which can be calculated as follows:
P(X<2) = P(X=0) + P(X=1)
= (10 choose 0)(0.1)^0(0.9)^10 + (10 choose 1)(0.1)^1(0.9)^9
= 0.3487 + 0.3874
= 0.736
Therefore, the statement is true and the answer is A.

Question 8

If X has a binomial distribution with n = 4 and p = 0.3, then P(X = 1) = .

Accepted Answer

The answer of If X has a binomial distribution with...

Question 9

SCENARIO 5-2
A certain type of new business succeeds 60% of the time.Suppose that 3 such businesses open (where they do not compete, so it is reasonable to believe that their relative successes would be independent).
-Referring to Scenario 5-2, the probability that all 3 businesses fail is _.

Accepted Answer

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

Question 10

Suppose that the number of airplanes arriving at an airport per minute is a Poisson process.The mean 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 probability of observing exactly $k$ events in a fixed interval in a Poisson process is given by $P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!}$, where $\lambda$ is the average rate (mean) of occurrences in that interval, and $k$ is the number of occurrences. For $\lambda = 3$ and $k = 6$, the formula yields $P(6; 3) = \frac{e^{-3} 3^6}{6!} \approx 0.0504$, matching the provided probability.

Question 11

SCENARIO 5-2
A certain type of new business succeeds 60% of the time.Suppose that 3 such businesses open (where they do not compete, so it is reasonable to believe that their relative successes would be independent).
-Referring to Scenario 5-2, the probability that exactly 1 business succeeds is .

Accepted Answer

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

Question 12

SCENARIO 5-1
The probability that a smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8.You have 2 such alarms in your home and they operate independently.
-Referring to Scenario 5-1, the probability that both sound an alarm in the presence of smoke is_.

Accepted Answer

The answer of SCENARIO 5-1
The probability that a smoke alarm...

Question 13

SCENARIO 5-1
The probability that a smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8.You have 2 such alarms in your home and they operate independently.
-Referring to Scenario 5-1, the probability that at least one sounds an alarm in the presence of smoke is _.

Accepted Answer

The answer of SCENARIO 5-1
The probability that a smoke alarm...

Question 14

The number of customers arriving at a department store in a 5-minute period has aPoisson distribution.

Accepted Answer

The Poisson distribution is commonly used to describe the number of events that occur in a fixed time or space interval, such as the number of customers arriving at a department store in a 5-minute period. Therefore, the statement is true.

Question 15

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

Accepted Answer

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

Question 16

If  $\pi$ remains constant in a binomial distribution, an increase in n will increase the variance.

Accepted Answer

This statement is true according to the binomial distribution formula, which states that the variance of a binomial distribution is equal to n*p*(1-p), where n is the number of trials and p is the probability of success on each trial. Since F1F1F1S1F1F1F10 represents the probability of success, if it remains constant and n increases, the variance will also increase.

Question 17

SCENARIO 5-2
A certain type of new business succeeds 60% of the time.Suppose that 3 such businesses open (where they do not compete, so it is reasonable to believe that their relative successes would be independent).
-Referring to Scenario 5-2, the probability that all 3 businesses succeed is .

Accepted Answer

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

Question 18

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

Accepted Answer

This is true. The expected value is calculated as the sum of each possible outcome multiplied by its probability, which is the same as the calculation for the mean.

Question 19

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 mean of a Poisson distribution represents the average or expected number of events occurring in a given time period. In this case, the mean of 6.5 tickets per day means that if we sampled all days, we would expect the average number of tickets written to be 6.5 per day.

Question 20

SCENARIO 5-2
A certain type of new business succeeds 60% of the time.Suppose that 3 such businesses open (where they do not compete, so it is reasonable to believe that their relative successes would be independent).
-Referring to Scenario 5-2, the probability that at least 1 business succeeds is .

Accepted Answer

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

Quiz 5: Discrete Probability Distributions

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

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

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

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

SCENARIO 5-1 The probability that a smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8.You have 2 such alarms in your home and they operate independently. -Referring to Scenario 5-1, the probability that neither sound an alarm in the presence of smoke is_.

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

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.

If X has a binomial distribution with n = 4 and p = 0.3, then P(X = 1) = .

SCENARIO 5-2 A certain type of new business succeeds 60% of the time.Suppose that 3 such businesses open (where they do not compete, so it is reasonable to believe that their relative successes would be independent). -Referring to Scenario 5-2, the probability that all 3 businesses fail is _.

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

SCENARIO 5-2 A certain type of new business succeeds 60% of the time.Suppose that 3 such businesses open (where they do not compete, so it is reasonable to believe that their relative successes would be independent). -Referring to Scenario 5-2, the probability that exactly 1 business succeeds is .

SCENARIO 5-1 The probability that a smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8.You have 2 such alarms in your home and they operate independently. -Referring to Scenario 5-1, the probability that both sound an alarm in the presence of smoke is_.

SCENARIO 5-1 The probability that a smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8.You have 2 such alarms in your home and they operate independently. -Referring to Scenario 5-1, the probability that at least one sounds an alarm in the presence of smoke is _.

The number of customers arriving at a department store in a 5-minute period has aPoisson distribution.

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

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

SCENARIO 5-2 A certain type of new business succeeds 60% of the time.Suppose that 3 such businesses open (where they do not compete, so it is reasonable to believe that their relative successes would be independent). -Referring to Scenario 5-2, the probability that all 3 businesses succeed is .

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

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.

SCENARIO 5-2 A certain type of new business succeeds 60% of the time.Suppose that 3 such businesses open (where they do not compete, so it is reasonable to believe that their relative successes would be independent). -Referring to Scenario 5-2, the probability that at least 1 business succeeds is .

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

If $\pi$ remains constant in a binomial distribution, an increase in n will increase the variance.