Question 1

Which of the following scenarios is best modeled by a Poisson random variable?

Accepted Answer

Counting the number of times a person sneezes in a week.

Question 2

Suppose customers arrive at a coffee shop at a rate of 2 per 15 minutes. What is the expected number of customers in 45 minutes?

Accepted Answer

$$6$$

Question 3

The probability mass function for a Poisson random variable $$X$$ with rate $$\lambda is $$given by:

Accepted Answer

$$P(X = x) = \dfrac{e$$^{-\lambda}$$ \lambda^x}{x!}$$

Question 4

If $$X \sim 	ext{Pois}(\lambda)$$, what are the mean and variance of $$X$$?

Accepted Answer

Mean: $$\lambda$$, Variance: $$\lambda$$

Question 5

A rare disease affects 1 in 100,000 people. In a university of 35,821 students, what is the value of $$\lambda$$ for the Poisson approximation?

Accepted Answer

$$\lambda = 0.35821$$

Question 6

Which condition justifies using the Poisson approximation to the Binomial distribution?

Accepted Answer

When $$n is $$large and $$p is $$small.

Question 7

Suppose a store receives customers at a rate of 1 per 10 minutes. What is the probability of receiving exactly 3 customers in 30 minutes?

Accepted Answer

$$P(X=3) = \dfrac{e$$^{-3}$$ 3^3$}{6}$$

Question 8

Wayne Gretzky scored 1669 points in 696 games. If the number of points per game is modeled as $$X \sim 	ext{Pois}(2.398)$$, what is the expected number of games with no points?

Accepted Answer

$$696 	imes 0.091$$

Question 9

A batch of 7000 items has a defect rate of 1 in 5000. Using the Poisson approximation, what is the expected number of defective items?

Accepted Answer

$$\lambda = 1.4$$

Question 10

Which of the following is NOT a key property of the Poisson process?

Accepted Answer

Events can occur simultaneously.

Question 11

Suppose fatal car accidents in a town happen at a rate of 1.5 per month. What is the probability of having at most one fatal accident in two months?

Accepted Answer

$$P(X \leq 1) = e$$^{-3}$$ + 3e$$^{-3}$$

Question 12

Which random variable counts the number of trials until the first success?

Accepted Answer

Geometric random variable

Question 13

If $$X \sim 	ext{Pois}(\lambda)$$, what is the standard deviation of $$X$$?

Accepted Answer

$$\sqrt{\lambda}$$

Question 14

Which of the following is the correct support for a Poisson random variable?

Accepted Answer

$$\{0, 1, 2, \ldots\}$$

Question 15

A student asks questions during a lecture at an average rate of 2 per hour. What is the probability that the student asks no questions in a 1-hour lecture?

Accepted Answer

$$e^{-2}$$

Question 16

Which statement best describes the difference between Binomial and Poisson random variables?

Accepted Answer

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

Question 17

If $$X \sim 	ext{Pois}(\lambda)$$, which of the following is true?

Accepted Answer

The mean and variance of $$X$$ are equal.

Question 18

A phone receives calls at a rate of 5 per hour. What is the probability of receiving exactly 0 calls in 1 hour?

Accepted Answer

$$e^{-5}$$

Question 19

Which of the following is a real-life example where the Poisson distribution is appropriate?

Accepted Answer

Number of emails received per hour.

Question 20

Suppose $$X \sim 	ext{Pois}(4). $$What is $$P(X = 2)$$?

Accepted Answer

$$\dfrac{e$$^{-4}$$ 4^2$}{2!}$$

Poisson Random Variable and Related Discrete Distributions

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