Question 1

The Poisson distribution is appropriate to describe the number of babies born in a small hospital on a given day.

Accepted Answer

The Poisson distribution is appropriate to describe the number of rare events occurring in a fixed time or space, such as the number of accidents per day on a certain road or the number of phone calls received by a call center in an hour. The number of babies born in a small hospital on a given day can also be considered as a rare event, and therefore the Poisson distribution is an appropriate model to describe it.

Question 2

A discrete random variable has a countable number of distinct values.

Accepted Answer

For example, the number of heads in three flips of a fair coin is a discrete random variable that can take on the values 0, 1, 2, or 3, all of which are countable and distinct.

Question 3

To describe the number of blemishes per sheet of white bond paper, we would use a discrete uniform distribution.

Accepted Answer

The number of blemishes per sheet of white bond paper would be described by a discrete distribution, but not necessarily a uniform distribution, because the probability of each possible number of blemishes occurring is not necessarily the same.

Question 4

The hypergeometric distribution is skewed right.

Accepted Answer

The skewness of a hypergeometric distribution can be either right, left, or symmetric depending on the parameters of the distribution (population size, number of successes in the population, and sample size).

Question 5

Calculating the probability of getting three aces in a hand of five cards dealt from a deck of 52 cards would require the use of a hypergeometric distribution.

Accepted Answer

The hypergeometric distribution is used to calculate the probability of getting a certain number of successes in a fixed number of draws without replacement from a finite population. In this case, we want to calculate the probability of getting three aces in a hand of five cards, which involves drawing without replacement from a finite population (the remaining 49 cards after the aces are removed). Therefore, the hypergeometric distribution is the appropriate probability distribution to use.

Question 6

Although the shape of the Poisson distribution is positively skewed, it becomes more nearly symmetric as its mean becomes larger.

Accepted Answer

This statement is true. As the mean of the Poisson distribution increases, the graph becomes more bell-shaped, and the skewness decreases.

Question 7

The hypergeometric distribution assumes that the probability of a success remains the same from one trial to the next.

Accepted Answer

The hypergeometric distribution models the probability of k successes in n draws without replacement from a finite population, which means the probability of success changes on each draw, unlike the binomial distribution where the probability of success remains the same on each trial.

Question 8

A random variable is a function or rule that assigns a numerical value to each outcome in the sample space of a stochastic (chance) experiment.

Accepted Answer

This is the definition of a random variable.

Question 9

A discrete binomial distribution is skewed right when π > .50.

Accepted Answer

A discrete binomial distribution is skewed left when π > .50, as the probability of success increases, making the tail on the left side longer.

Question 10

The Poisson distribution describes the number of occurrences within a randomly chosen unit of time or space.

Accepted Answer

The Poisson distribution is used to describe the number of occurrences in a fixed interval of time or space.

Question 11

The outcomes for the sum of two dice can be described as a discrete uniform distribution.

Accepted Answer

The outcomes for the sum of two dice is not a discrete uniform distribution as not all outcomes have the same probability. The outcomes 2 and 12 have only one possible combination each (1+1 and 6+6) while 7 has the highest probability with six possible combinations (1+6, 2+5, 3+4, 4+3, 5+2, 6+1).

Question 12

The hypergeometric distribution is not applicable if sampling is done with replacement.

Accepted Answer

The hypergeometric distribution assumes that sampling is done without replacement, where the probability of each outcome is dependent on the previous selections. If sampling is done with replacement, the probability of each outcome remains the same regardless of previous selections, and the distribution would be better modeled by the binomial distribution.

Question 13

An example of a geometric random variable is the number of pine trees with pine beetle infestation in a random sample of 15 pine trees in Colorado.

Accepted Answer

A geometric random variable counts the number of trials until the first success in a series of independent Bernoulli trials, not the total number of successes in a fixed number of trials.

Question 14

A random variable may be discrete or continuous, but not both.

Accepted Answer

A random variable is classified as either discrete, taking on a countable number of distinct values, or continuous, taking on an uncountable range of values, but it cannot be both simultaneously.

Question 15

As a rule of thumb, the Poisson distribution can be used to approximate a binomial distribution when n ≥ 20 and π ≤ .05.

Accepted Answer

This statement is true. When n is large and π is small, calculating the probabilities for each possible outcome in a binomial distribution can become very time-consuming. In these cases, a Poisson distribution can be used as an approximation. The rule of thumb is n ≥ 20 and π ≤ .05, but the exact criteria may vary depending on the specific situation.

Question 16

A discrete distribution can be described by its probability density function (PDF) or by its cumulative distribution function (CDF).

Accepted Answer

A discrete distribution can be described by its probability density function (PDF) or by its cumulative distribution function (CDF).

Question 17

The Poisson distribution can be skewed either left or right, depending on λ.

Accepted Answer

The Poisson distribution is a discrete probability distribution often used to model the number of occurrences of rare events in a fixed amount of time or space. The shape of the distribution is determined solely by its parameter λ, also known as the mean or expected value of the distribution. The Poisson distribution is always right-skewed, meaning there is a long tail in the positive direction. It cannot be left-skewed.

Question 18

When π = .70 the discrete binomial distribution is negatively skewed.

Accepted Answer

When the probability of success (π) is less than .50, the binomial distribution is negatively skewed. When π is greater than .50, the distribution is positively skewed. When π is exactly .50, the distribution is symmetrical.

Question 19

The expected value of a discrete random variable E(X) is the sum of all X values weighted by their respective probabilities.

Accepted Answer

This is a definition of the expected value of a discrete random variable, which is defined as the sum of all X values weighted by their respective probabilities.

Question 20

As a rule of thumb, the binomial distribution can be used to approximate the hypergeometric distribution whenever the population is at least 20 times as large as the sample.

Accepted Answer

This is a commonly used approximation rule in statistics. When the population is much larger than the sample, using a binomial distribution instead of a hypergeometric distribution can simplify calculations without significantly affecting the accuracy of the results.

Quiz 6: Discrete Probability Distributions

The Poisson distribution is appropriate to describe the number of babies born in a small hospital on a given day.

A discrete random variable has a countable number of distinct values.

To describe the number of blemishes per sheet of white bond paper, we would use a discrete uniform distribution.

The hypergeometric distribution is skewed right.

Calculating the probability of getting three aces in a hand of five cards dealt from a deck of 52 cards would require the use of a hypergeometric distribution.

Although the shape of the Poisson distribution is positively skewed, it becomes more nearly symmetric as its mean becomes larger.

The hypergeometric distribution assumes that the probability of a success remains the same from one trial to the next.

A random variable is a function or rule that assigns a numerical value to each outcome in the sample space of a stochastic (chance) experiment.

A discrete binomial distribution is skewed right when π > .50.

The Poisson distribution describes the number of occurrences within a randomly chosen unit of time or space.

The outcomes for the sum of two dice can be described as a discrete uniform distribution.

The hypergeometric distribution is not applicable if sampling is done with replacement.

An example of a geometric random variable is the number of pine trees with pine beetle infestation in a random sample of 15 pine trees in Colorado.

A random variable may be discrete or continuous, but not both.

As a rule of thumb, the Poisson distribution can be used to approximate a binomial distribution when n ≥ 20 and π ≤ .05.

A discrete distribution can be described by its probability density function (PDF) or by its cumulative distribution function (CDF).

The Poisson distribution can be skewed either left or right, depending on λ.

When π = .70 the discrete binomial distribution is negatively skewed.

The expected value of a discrete random variable E(X) is the sum of all X values weighted by their respective probabilities.

As a rule of thumb, the binomial distribution can be used to approximate the hypergeometric distribution whenever the population is at least 20 times as large as the sample.