Question 1

A college has seven applicants for three scholarships: four females and three males. Suppose that the seven applicants are equally qualified and that no preference is given by the selection committee for choosing either gender. Let x equal the number of female students chosen for the three scholarships.
What is the mean of the distribution of x?
______________
What is the variance of the distribution of x?
______________
What is the probability that only one female will receive a scholarship?
______________
What is the probability that two females will receive a scholarship?
______________
What is the probability that none of the three males will receive a scholarship?
______________
What is the probability that none of the four females will receive a scholarship?
______________

Accepted Answer

The answer of A college has seven applicants for three...

Question 2

Which of the following cannot generate a Poisson distribution?

Accepted Answer

The number of children in a family cannot generate a Poisson distribution as it cannot be counted as random events occurring in a specified time period, like the other options.

Question 3

Poisson distribution is appropriate to determine the probability of a given number of defective items in a shipment.

Accepted Answer

The Poisson distribution is used for modeling the number of events in a fixed interval of time or space, not for defective items in a shipment, which is better modeled by the binomial distribution.

Question 4

The mean and variance of the Poisson distribution are equal.

Accepted Answer

The mean and variance of the Poisson distribution are both equal to lambda, which is the rate parameter of the distribution. Therefore, the statement is true.

Question 5

In a Poisson problem, x represents the number of events occurring in a period of time or space during which an average of  such events can be expected to occur.

Accepted Answer

This is the definition of a Poisson distribution - the number of events occurring in a period of time or space during which an average of λ such events can be expected to occur. The given value of λ is 11eb6b9e_0668_fa67_8d9d_67bac4d2f07c_TB8220_11.

Question 6

The Poisson parameter  is the mean number of occurrences of an event per unit of time or space during the Poisson process.

Accepted Answer

The Poisson parameter is defined as the average number of occurrences of an event in a given time or space interval in a Poisson process.

Question 7

The Poisson probability distribution provides good approximations to binomial probabilities when n is large and  is small, preferably with np < 7.

Accepted Answer

This is a well-known fact in probability theory. When n is large and 11eb6b9e_0668_fa68_8d9d_fb4dd19ec637_TB8220_11 is small, the binomial distribution can be approximated by the Poisson distribution with λ=np.

Question 8

The Poisson random variable is:

Accepted Answer

The Poisson random variable is a discrete random variable with infinitely many possible values, where these values represent the number of occurrences of a rare event in a given time or space interval. It is not a continuous random variable, nor does it have a finite number of possible values. Therefore, the best choice is B.

Question 9

Given a Poisson random variable x, where the average number of times an event occurs in a certain period of time is 2.5, then P(x = 0) is:

Accepted Answer

Using the Poisson formula, P(x = k) = (λ^k * e^(-λ)) / k!, where λ is the average number of events and k is the specific number of events we want to find the probability of. 
In this case, λ = 2.5 and k = 0, so P(x = 0) = (2.5^0 * e^(-2.5)) / 0! = 0.0821.

Question 10

A Poisson process is the occurrence of a series of events of a given type in a random pattern over time or space such that (1) the number of occurrences within a specified time or space can equal any integer between zero and infinity, (2) the number of occurrences within one unit of time or space is independent of that in any other such (non-overlapping) unit, and (3) the probability of occurrences is the same in all such units.

Accepted Answer

This is the definition of a Poisson process.

Question 11

The Poisson probability distribution is an example of continuous probability distribution.

Accepted Answer

The Poisson distribution is an example of a discrete probability distribution.

Question 12

Which probability distribution is appropriate when the events of interest occur randomly, independently of one another, and rarely?

Accepted Answer

The Poisson distribution is appropriate when events occur randomly and independently of one another, but do so rarely. The binomial distribution is appropriate when events have a fixed number of trials and a fixed probability of success, while the hypergeometric distribution is appropriate when events are selected without replacement from a finite population. Therefore, option B is the best choice in this scenario.

Question 13

The Poisson probability distribution is an example of a continuous probability distribution.

Accepted Answer

The Poisson probability distribution is an example of a discrete probability distribution, as it deals with the number of occurrences of a random event in a fixed interval of time or space, and the values it takes are whole numbers.

Question 14

The mean of a Poisson distribution, where  is the average number of successes occurring in a specified interval, is  .

Accepted Answer

The answer of The mean of a Poisson distribution, where...

Question 15

The Poisson probability tables list the probabilities of x occurrences in a Poisson process for various values of  , the mean number of occurrences.

Accepted Answer

The Poisson probability tables do indeed list the probabilities of x occurrences in a Poisson process for various values of the mean, λ.

Question 16

The Poisson random variable is the number of successes achieved when a random sample of size n is drawn without replacement from a population of size N within which M units have the characteristic that denotes success.

Accepted Answer

The Poisson random variable is defined as the number of occurrences of a rare event in a fixed amount of time or space, with an assumed constant rate of occurrence. It does not depend on a sample size or population size, and it assumes the events are independent and randomly distributed in time or space. The scenario described in the statement is a hypergeometric distribution, which does depend on sample size and population size, and it assumes sampling without replacement.

Question 17

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.

Accepted Answer

The Poisson distribution is used for events where the probability of occurrence over a given time, space, or distance is small.

Question 18

The Poisson distribution is applied to events for which the probability of occurrence over a given span of time, space, or distance is large.

Accepted Answer

The Poisson distribution is applied to model the number of times an event occurs in a fixed interval of time or space when these events happen with a known constant mean rate and independently of the time since the last event. It is typically used for rare events.

Question 19

The probability distribution of a Poisson random variable provides a good model for data that represent the number of occurrences of a specified event in a given unit of time or space.

Accepted Answer

The Poisson distribution is commonly used to model count data, such as the number of earthquakes in a region or the number of customers who enter a store. It assumes that the occurrences are random and independent, and that the average rate of occurrences is constant over time or space. Therefore, it is a suitable model for data that meet these assumptions.

Question 20

Which of the following experiments can be modeled by the Poisson distribution?

Accepted Answer

All of these experiments can be modeled by the Poisson distribution, as they involve counting the number of occurrences of a rare event in a given time or space. The Poisson distribution is commonly used to model the number of rare events, such as the number of calls received by a switchboard, the number of bacteria in a sample, or the number of customer arrivals at a checkout counter, as long as the events occur independently of each other and with a fixed mean rate.

Quiz 5: Several Useful Discrete Distributions