PMF Geometric Distribution 2 — Questions and Answers
Question 1: For a Geometric distribution with p = 0.25, what is P(X = 3)?
- (0.75)² × 0.25 ≈ 0.141 (Correct answer)
- (0.25)² × 0.75 ≈ 0.047
- 0.75 × 0.25 ≈ 0.188
- (0.25)³ ≈ 0.016
Correct answer: (0.75)² × 0.25 ≈ 0.141
P(X=3) = (1-0.25)^(3-1) × 0.25 = (0.75)² × 0.25 = 0.5625 × 0.25 ≈ 0.141.
Question 2: What is the support of the Geometric distribution (number-of-trials version)?
- {1, 2, 3, 4, …} (Correct answer)
- {0, 1, 2, 3, …}
- {1, 2, …, n}
- {0, 1}
Correct answer: {1, 2, 3, 4, …}
Since at least one trial is always needed, the support starts at 1 and extends to infinity.
Question 3: Using the memoryless property, if X ~ Geometric(p=0.3) and you have already failed 5 times, what is the probability the next trial succeeds?
- 0.3 (Correct answer)
- 0.7
- 0.3 × (0.7)^5
- 1 - (0.7)^5
Correct answer: 0.3
Due to the memoryless property, the probability of success on the next trial is always p = 0.3 regardless of past failures.
Question 4: How does increasing p affect the Geometric PMF?
- The distribution becomes more concentrated near k=1 (Correct answer)
- The distribution becomes more spread out
- The mean increases
- The variance increases
Correct answer: The distribution becomes more concentrated near k=1
Higher p means success is more likely quickly, so the distribution concentrates near smaller values of k.
Question 5: What is the CDF of the Geometric distribution, P(X ≤ k)?
- 1 - (1-p)^k (Correct answer)
- (1-p)^k
- 1 - p^k
- p^k
Correct answer: 1 - (1-p)^k
P(X ≤ k) = 1 - (1-p)^k, representing the probability that the first success occurs by the k-th trial.
Question 6: Which scenario is best modeled by a Geometric distribution?
- Rolling a die until getting a 6 (Correct answer)
- Counting the number of 6s in 10 rolls
- Time between two bus arrivals
- Number of heads in 20 coin flips
Correct answer: Rolling a die until getting a 6
Rolling a die until getting a 6 counts the number of trials to first success, which is exactly what the Geometric distribution models.
PMF Geometric Distribution 2
For a Geometric distribution with p = 0.25, what is P(X = 3)?