Question 1

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

Accepted Answer

The number of customers arriving at a store in one hour.

Question 2

Suppose X is a Poisson random variable with rate $$\lambda = 3. $$What is $$P(X = 2)$$?

Accepted Answer

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

Question 3

If the average rate of car accidents in a city is 1.5 per month, what is the expected number of accidents in 4 months?

Accepted Answer

$$6$$

Question 4

For a Poisson random variable $$X \sim Pois(\lambda)$$, which of the following is true?

Accepted Answer

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

Question 5

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

Accepted Answer

$$0.35821$$

Question 6

If $$X \sim Pois(2.398)$$, what is the probability that $$X = 0$$?

Accepted Answer

$$e^{-2.398}$$

Question 7

Which condition must be met to use the Poisson approximation for a Binomial distribution?

Accepted Answer

The number of trials $$n is $$large and the probability of success $$p is $$small.

Question 8

A store receives customers at a rate of 1 per 10 minutes. What is the probability of receiving exactly 4 customers in 30 minutes?

Accepted Answer

$$\frac{e$$^{-3}$$ \cdot 3^4$}{4!}$$

Question 9

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

Accepted Answer

$$\sqrt{\lambda}$$

Question 10

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

$$1.4$$

Question 11

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

Accepted Answer

Events can occur simultaneously.

Question 12

Wayne Gretzky scored an average of 2.398 points per game. What is the probability he scores exactly 1 point in a game?

Accepted Answer

$$e^{-2.398}$$ \cdot \frac{$$2.398^1$$}{1!}$$

Question 13

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

Accepted Answer

Geometric random variable

Question 14

Suppose a phenomenon occurs at a rate of $$\lambda = 5$$ per hour. What is the probability that no events occur in one hour?

Accepted Answer

$$e^{-5}$$

Question 15

If $$X \sim Pois(\lambda)$$, which set represents the support of $$X$$?

Accepted Answer

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

Question 16

A student asks questions during a lecture at an average rate of 2 per hour. What is the probability that the student asks at least 3 questions in one hour?

Accepted Answer

$$1 - \left( e$$^{-2}$$ + 2e$$^{-2}$$ + 2^2$ e$$^{-2}$$/2! \right)$$

Question 17

Which of the following is a correct application of the Poisson approximation to the Binomial distribution?

Accepted Answer

When $$n is $$large and $$p is $$small, use $$\lambda = np in $$the Poisson formula.

Question 18

Which formula gives the probability that a Poisson random variable $$X$$ with rate $$\lambda$$ equals $$x$$?

Accepted Answer

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

Poisson Random Variable and Related Discrete Distributions

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