Question 1

Which of the following scenarios is most appropriately modeled by a Poisson distribution?

Accepted Answer

The number of emails received per hour at a help desk.

Question 2

Suppose a call center receives an average of 3 calls per hour. What is the probability that exactly 5 calls are received in one hour? Use the Poisson formula.

Accepted Answer

$$\frac{3^5$ e$$^{-3}$$}{5!}$$

Question 3

A machine breaks down on average once every two weeks. What is the expected number of breakdowns in the next six weeks?

Accepted Answer

$$3$$

Question 4

Which of the following is NOT an assumption of the Poisson distribution?

Accepted Answer

Two events can occur at exactly the same instant.

Question 5

If $$X \sim 	ext{Poisson}(\lambda)$$, what is the variance of $$X$$?

Accepted Answer

$$\lambda$$

Question 6

A publishing company prints 450,000 letters in a novel. If the probability of a typo for any given letter is $$0.00001$$, what is the expected number of typos in the novel?

Accepted Answer

$$4.5$$

Question 7

Which formula gives the probability mass function (PMF) for the Poisson distribution?

Accepted Answer

$$p(x) = \frac{\lambda^x e$$^{-\lambda}$$}{x!}$$ for $$x = 0, 1, 2, \ldots$$

Question 8

Under what conditions can a binomial distribution be approximated by a Poisson distribution?

Accepted Answer

$$n is $$large, $$p is $$small, and $$np is $$moderate (typically $$n \geq 100$$ and $$np \leq 10)$$

Question 9

A bank receives on average 0.5 customer arrivals per minute. What is the probability that no customers arrive in a given minute?

Accepted Answer

$$e^{-0.5}$$

Question 10

If $$X \sim 	ext{Poisson}(\lambda)$$, which of the following statements is TRUE?

Accepted Answer

The mean and variance of $$X$$ are both equal to $$\lambda.$$

Question 11

A machine breaks down on average once every two weeks. What is the probability that it breaks down exactly three times in the next four weeks?

Accepted Answer

$$\frac{2^3$ e$$^{-2}$$}{3!}$$

Question 12

Which of the following is a key difference between the Poisson and Binomial distributions?

Accepted Answer

Poisson counts events in a fixed interval; Binomial counts successes in a fixed number of trials.

Question 13

If the average number of emails received per hour is $$0.5$$, what is the expected number of emails received per day?

Accepted Answer

$$12$$

Question 14

In the study of bomb hits in London during WWII, 537 bombs struck an area of 144 km$$^2$$ divided into 576 squares. What is the average number of bombs per square?

Accepted Answer

$$0.93$$

Question 15

Which of the following best describes the type of random variable modeled by the Poisson distribution?

Accepted Answer

The number of events occurring in a fixed interval of time or space.

Question 16

A website receives on average 0.0083 clicks per minute. What is the expected number of clicks in an hour?

Accepted Answer

$$0.5$$

Question 17

Which of the following is a valid reason to use the Poisson approximation to the Binomial distribution?

Accepted Answer

When $$n is $$very large and $$p is $$very small, making direct Binomial calculations difficult.

Question 18

If $$X \sim 	ext{Poisson}(\lambda)$$, what is the probability that $$X = 0$$?

Accepted Answer

$$e^{-\lambda}$$

Question 19

A Poisson process is observed for 10 hours, with an average rate of 2 events per hour. What is the probability of observing exactly 20 events in 10 hours?

Accepted Answer

$$\frac{20$$^{20}$$ e$$^{-20}$$}{20!}$$

5.2 Poisson Distribution: Properties, Applications, and Approximations

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