PMF Poisson Distribution 3

Question 1

Which real-world scenario is best modeled by a Poisson PMF?

Accepted Answer

Number of typos per page in a manuscript

Answer

Rare, independent events per unit (like typos per page) follow a Poisson distribution.

Question 2

If the Poisson parameter λ increases, how does the shape of the PMF change?

Accepted Answer

It shifts right and becomes more symmetric

Answer

As λ increases, the Poisson distribution shifts right and its shape approaches a symmetric bell curve.

Question 3

What is the mode of a Poisson distribution when λ = 5.5?

Accepted Answer

Both 5 and 6

Answer

When λ is not an integer, the Poisson distribution has two modes at ⌊λ⌋ and ⌈λ⌉ - 1, which are both 5 here. Actually when λ is not an integer, the unique mode is ⌊λ⌋ = 5; when λ is a positive integer, modes are at λ-1 and λ.

Question 4

For a Poisson PMF with λ = 1, which value of k gives the highest probability?

Accepted Answer

0

Answer

P(X=0) = e^-1 ≈ 0.368 and P(X=1) = e^-1 ≈ 0.368 are equal when λ=1, but k=0 is also a mode.

Question 5

How does the standard deviation of a Poisson(λ=9) compare to its mean?

Accepted Answer

The standard deviation (3) equals the square root of the mean (9)

Answer

For Poisson, Var(X) = λ = 9, so SD = √9 = 3, which is the square root of the mean.

PMF - Probability Mass Function Practice Test

PMF - Probability Mass Function Practice Test

PMF Poisson Distribution 3