Q: If $$X is a $$Poisson random variable with mean $$\mu = 3$$ for one hour, what is the mean number of occurrences in 4 hours?

$$\mu_{4\text{ hours}} = 4 \times 3 = 12$$

Question 1

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

Accepted Answer

The occurrences must be random and independent of each other.

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 a given hour? Use the Poisson formula $$P(x) = \frac{\mu^x \cdot e$$^{-\mu}$$}{x!}$$ with $$\mu = 3$$ and $$x = 5.$$

Accepted Answer

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

Question 3

If the mean number of car accidents at a certain intersection is 2 per month, what is the standard deviation of the number of accidents per month?

Accepted Answer

$$\sigma = \sqrt{2} \approx 1.414$$

Question 4

A Poisson distribution is most appropriate for modeling which of the following scenarios?

Accepted Answer

The number of emails received by a company in an hour.

Question 5

Given a Poisson process with a mean of $$\mu = 4$$ events per day, what is the probability of observing zero events in a day?

Accepted Answer

$$P(0) = \frac{4^0$ \cdot e$$^{-4}$$}{0!} = e$$^{-4}$$ \approx 0.0183$$

Question 6

If the mean number of hurricanes per year is $$\mu = 5.5$$, what is the probability of exactly 6 hurricanes in a year? (Use $$e = 2.71828$$ and $$6! = 720)$$

Accepted Answer

$$P(6) = \frac{5.5^6$ \cdot e$$^{-5.5}$$}{720} \approx 0.157$$

Question 7

Which of the following statements about the Poisson distribution is FALSE?

Accepted Answer

The Poisson distribution has a fixed upper limit for $$x.$$

Question 8

A hospital records an average of 2 births per hour. What is the probability that exactly 3 births occur in a given hour?

Accepted Answer

$$P(3) = \frac{2^3$ \cdot e$$^{-2}$$}{3!} \approx 0.180$$

Question 9

If $$X is a $$Poisson random variable with mean $$\mu = 3$$ for one hour, what is the mean number of occurrences in 4 hours?

Accepted Answer

$$\mu_{4	ext{ hours}} = 4 	imes 3 = 12$$

Question 10

A rare disease occurs on average 0.2 times per 100,000 people per year. What is the probability that in a city of 100,000 people, there are no cases in a year?

Accepted Answer

$$P(0) = e$$^{-0.2}$$ \approx 0.8187$$

Question 11

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

Accepted Answer

The probability of each event is dependent on the previous event.

Question 12

A website receives an average of 10 hits per minute. What is the probability that it receives exactly 8 hits in a minute?

Accepted Answer

$$P(8) = \frac{10^8$ \cdot e$$^{-10}$$}{8!} \approx 0.1126$$

Question 13

If the mean number of emails received per hour is 6, what is the probability of receiving at least one email in an hour?

Accepted Answer

$$P(	ext{at least 1}) = 1 - e$$^{-6}$$ \approx 0.9975$$

Question 14

A Poisson process has a mean of $$\mu = 2$$ per hour. What is the probability of observing 4 events in 2 hours?

Accepted Answer

$$P(4; \mu=4) = \frac{4^4$ \cdot e$$^{-4}$$}{4!} \approx 0.1954$$

Question 15

Which of the following best describes the parameter $$\mu in $$the Poisson distribution?

Accepted Answer

It is the mean number of occurrences of the event in the specified interval.

Question 16

A researcher counts the number of bacteria colonies on a petri dish and finds an average of 7 colonies per dish. What is the probability of finding exactly 10 colonies on a dish?

Accepted Answer

$$P(10) = \frac{7$$^{10}$$ \cdot e$$^{-7}$$}{10!} \approx 0.0712$$

Question 17

If the mean number of earthquakes in a region is 1.5 per year, what is the probability that there are no earthquakes in a given year?

Accepted Answer

$$P(0) = e$$^{-1.5}$$ \approx 0.2231$$

Question 18

A Poisson distribution is used to model the number of flaws in a 10-meter length of wire, with a mean of 0.8 flaws per 10 meters. What is the probability of finding exactly 2 flaws in a 10-meter length?

Accepted Answer

$$P(2) = \frac{0.8^2$ \cdot e$$^{-0.8}$$}{2!} \approx 0.1436$$

Poisson Probability Distributions: Concepts, Properties, and Applications

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