PMF Geometric Distribution 1 — Questions and Answers
Question 1: What does the Geometric distribution model?
- The number of trials until the first success (Correct answer)
- The total number of successes in n trials
- The number of failures before the k-th success
- The time between two Poisson events
Correct answer: The number of trials until the first success
The Geometric distribution models the number of independent Bernoulli trials needed to achieve the first success.
Question 2: What is the PMF of the Geometric distribution (number of trials until first success)?
- P(X=k) = (1-p)^(k-1) × p (Correct answer)
- P(X=k) = p^k × (1-p)
- P(X=k) = p^(k-1) × (1-p)
- P(X=k) = k × p × (1-p)^k
Correct answer: P(X=k) = (1-p)^(k-1) × p
The Geometric PMF P(X=k) = (1-p)^(k-1) × p represents k-1 failures followed by one success.
Question 3: What is the mean (expected value) of a Geometric distribution with success probability p?
- 1/p (Correct answer)
- p
- 1/(1-p)
- (1-p)/p
Correct answer: 1/p
The expected number of trials until the first success is E[X] = 1/p.
Question 4: A basketball player has a 40% free throw success rate. What is the expected number of throws to make the first basket?
- 2.5 (Correct answer)
- 0.4
- 4
- 1.6
Correct answer: 2.5
E[X] = 1/p = 1/0.4 = 2.5 throws expected until the first success.
Question 5: Which property makes the Geometric distribution unique among discrete distributions?
- Memoryless property (Correct answer)
- Symmetry around the mean
- Both mean and variance equal λ
- Sum of two Geometric variables is Geometric
Correct answer: Memoryless property
The Geometric distribution is the only discrete distribution with the memoryless property: past failures do not affect future probabilities.
Question 6: What is the variance of a Geometric distribution with parameter p?
- (1-p)/p² (Correct answer)
- 1/p
- p/(1-p)
- (1-p)/p
Correct answer: (1-p)/p²
The variance of a Geometric distribution is Var(X) = (1-p)/p².
PMF Geometric Distribution 1
What does the Geometric distribution model?