Question 1

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 Poisson process is given by $P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!}$, where $\lambda$ is the average rate of occurrence (mean) and $k$ is the number of occurrences. Here, with $\lambda = 3$ and $k = 6$, the formula yields approximately 0.0504, confirming the statement.

Question 2

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

Accepted Answer

A binomial distribution is used for counting the number of successes or failures in a fixed number of independent trials. The diameter of a bolt is a continuous variable and not a binary success/failure outcome, so it cannot have a binomial distribution.

Question 3

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...

Question 4

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

Accepted Answer

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

Question 5

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 value of the number of events occurring in a specific 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 around 6.5 tickets per day. This does not necessarily mean that 6.5 tickets will be written on any given day, but over time, the average number of tickets will approach 6.5.

Question 6

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

Accepted Answer

In a binomial distribution, the formula for variance is given by nπ(1-π). Since F1F1F1S1F1F1F10 remains constant, π is constant as well. Therefore, an increase in n will increase the variance.

Question 7

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 8

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 9

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

Accepted Answer

This is true for Poisson distribution, where the mean (λ) and variance (also λ) are equal.

Question 10

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 model the number of events that occur within a fixed time period, such as the number of customers arriving at a store in a certain time period. Therefore, the statement is true and the number of customers arriving in a 5-minute period can be modeled by a Poisson distribution.

Question 11

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 12

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 13

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 14

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 15

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

We can use the binomial distribution formula to solve this problem:
P(X < 2) = P(X = 0) + P(X = 1)
where X is the number of incorrect verdicts in 10 decisions.

The probability of an incorrect verdict is 0.1, so the probability of a correct verdict is 0.9. Using the binomial distribution formula, we have:

P(X = k) = (10 choose k) * (0.1)^k * (0.9)^(10-k)

Thus, 
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) = P(X = 0) + P(X = 1) = 0.3487 + 0.3874 = 0.7361

As the question states that the probability of fewer than 2 incorrect verdicts is 0.736, it is true that the judge's next 10 decisions will have a probability less than 2 incorrect verdicts of 0.736.

Question 16

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

Accepted Answer

A Poisson random variable can technically take on any non-negative value. However, as the mean of the Poisson distribution approaches infinity, the probability of the random variable taking on extremely large values approaches zero. Therefore, while there is no theoretical upper limit to the value that X can take, it becomes increasingly unlikely for X to take on extremely large values.

Question 17

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 18

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 19

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

Accepted Answer

The Poisson distribution is a discrete distribution used to model the number of occurrences of a rare event in a fixed interval of time or space. It cannot be used to model a continuous random variable.

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 all 3 businesses fail is _.

Accepted Answer

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

Quiz 5: Discrete Probability Distributions

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.

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

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 .

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

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.

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

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 exactly 1 business succeeds is .

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

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

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.

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.

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

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_.

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_.

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

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 _.

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

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