Q: In a 100-day period, a website receives an average of 2 complaints per day. What is the expected number of days with exactly 3 complaints?

100 $$\times$$ $$P(3)$$, where $$P(3) = \frac{2^3$ \cdot 2.71828$$^{-2}$$}{3!} = 0.1804$$, so expected = 18.04 days

Question 1

Which of the following is a requirement for a Poisson probability distribution?

Accepted Answer

The occurrences must be uniformly distributed over the interval.

Question 2

Given a Poisson distribution with a mean of $$\mu = 4$$, what is the probability of exactly 2 occurrences in an interval? Use $$e \approx 2.71828.$$

Accepted Answer

$$P(2) = \frac{4^2$ \cdot 2.71828$$^{-4}$$}{2!} = 0.1465$$

Question 3

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

Accepted Answer

The number of patients arriving at an emergency room in one hour.

Question 4

If the mean number of emails received per hour is $$\mu = 8$$, what is the standard deviation of the number of emails received per hour?

Accepted Answer

$$\sigma = \sqrt{8}$$

Question 5

A hospital records an average of 3 births per hour. What is the probability that exactly 5 births occur in a randomly selected hour? Use $$e \approx 2.71828.$$

Accepted Answer

$$P(5) = \frac{3^5$ \cdot 2.71828$$^{-3}$$}{5!} = 0.1008$$

Question 6

Which of the following is NOT a property of the Poisson distribution?

Accepted Answer

The random variable x must be continuous.

Question 7

In a 100-day period, a website receives an average of 2 complaints per day. What is the expected number of days with exactly 3 complaints?

Accepted Answer

100 $$	imes$$ $$P(3)$$, where $$P(3) = \frac{2^3$ \cdot 2.71828$$^{-2}$$}{3!} = 0.1804$$, so expected = 18.04 days

Question 8

When can the Poisson distribution be used as an approximation to the binomial distribution?

Accepted Answer

When $$n \geq 100$$ and $$np \leq 10$$

Question 9

If a Poisson distribution has $$\mu = 7$$, what is the probability of zero occurrences in an interval?

Accepted Answer

$$P(0) = 7^0$ \cdot 2.71828$$^{-7}$$ / 0! = 0.0009$$

Question 10

A call center receives an average of 12 calls per hour. What is the probability that exactly 10 calls are received in an hour? Use $$e \approx 2.71828.$$

Accepted Answer

$$P(10) = \frac{12$$^{10}$$ \cdot 2.71828$$^{-12}$$}{10!} = 0.1048$$

Question 11

Which formula correctly represents the Poisson probability of observing $$x$$ events in an interval with mean $$\mu$$?

Accepted Answer

$$P(x) = \frac{\mu^x \cdot e$$^{-\mu}$$}{x!}$$

Question 12

A rare disease occurs with a mean of 0.2 cases per year in a city. What is the probability that there will be at least one case in a year? Use $$e \approx 2.71828.$$

Accepted Answer

$$P(	ext{at least 1}) = 1 - P(0) = 1 - 2.71828$$^{-0.2}$$ = 0.1813$$

Question 13

If $$n = 200$$ and $$p = 0.03$$, can the Poisson distribution be used to approximate the binomial distribution? If so, what is $$\mu$$?

Accepted Answer

Yes, $$\mu = np = 200 	imes 0.03 = 6$$

Question 14

A traffic engineer observes an average of 4 car accidents per month at a certain intersection. What is the probability that there will be fewer than 2 accidents in a month? Use $$e \approx 2.71828.$$

Accepted Answer

$$P(0) + P(1) = \frac{4^0$ \cdot 2.71828$$^{-4}$$}{0!} + \frac{4^1$ \cdot 2.71828$$^{-4}$$}{1!} = 0.0183 + 0.0733 = 0.0916$$

Question 15

Which of the following is a real-world example where the Poisson distribution would NOT be appropriate?

Accepted Answer

The number of heads in 10 coin tosses.

5.3 Poisson Probability Distributions – Study Notes

Guida di studio - Domande di pratica

Study guide - Flashcard