PMF Poisson Distribution 2 — Questions and Answers
Question 1: The Poisson distribution is often used as an approximation for which distribution when n is large and p is small?
- Binomial distribution (Correct answer)
- Geometric distribution
- Hypergeometric distribution
- Uniform distribution
Correct answer: Binomial distribution
The Poisson distribution approximates the Binomial when n is very large and p is very small, using λ = np.
Question 2: For a Poisson PMF, which of the following is true about its support (possible values)?
- X can take any non-negative integer value (0, 1, 2, …) (Correct answer)
- X can only take values between 0 and 1
- X can take any positive real value
- X is limited to values from 0 to λ
Correct answer: X can take any non-negative integer value (0, 1, 2, …)
The support of a Poisson random variable is {0, 1, 2, 3, …}, all non-negative integers.
Question 3: If X ~ Poisson(λ=4), what is P(X=1)?
- 4e^-4 ≈ 0.073 (Correct answer)
- e^-4 ≈ 0.018
- 8e^-4 ≈ 0.147
- 4e^4
Correct answer: 4e^-4 ≈ 0.073
P(X=1) = (e^-4 × 4^1) / 1! = 4e^-4 ≈ 0.073.
Question 4: What is the sum of all probabilities P(X=k) for k=0,1,2,… in a Poisson distribution?
- 1 (Correct answer)
- λ
- e^λ
- e^-λ
Correct answer: 1
Like all valid PMFs, the Poisson probabilities sum to 1 over all possible values.
Question 5: Two independent Poisson random variables X ~ Poisson(λ₁) and Y ~ Poisson(λ₂) are added. What is the distribution of X + Y?
- Poisson(λ₁ + λ₂) (Correct answer)
- Poisson(λ₁ × λ₂)
- Poisson(λ₁ - λ₂)
- Normal(λ₁ + λ₂, λ₁ + λ₂)
Correct answer: Poisson(λ₁ + λ₂)
The sum of independent Poisson random variables is also Poisson with rate equal to the sum of the individual rates.
Question 6: In the Poisson PMF, what role does the constant e (Euler's number ≈ 2.718) play?
- It is part of the normalizing factor ensuring probabilities sum to 1 (Correct answer)
- It represents the average number of events
- It scales the number of trials
- It defines the upper bound of the distribution
Correct answer: It is part of the normalizing factor ensuring probabilities sum to 1
The e^-λ term in the Poisson PMF is the normalizing constant that ensures all probabilities sum to exactly 1.
PMF Poisson Distribution 2
The Poisson distribution is often used as an approximation for which distribution when n is large and p is small?