PMF Geometric Distribution 2

Question 1

For a Geometric distribution with p = 0.25, what is P(X = 3)?

Accepted Answer

(0.75)² × 0.25 ≈ 0.141

Answer

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)?

Accepted Answer

{1, 2, 3, 4, …}

Answer

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?

Accepted Answer

0.3

Answer

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?

Accepted Answer

The distribution becomes more concentrated near k=1

Answer

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)?

Accepted Answer

1 - (1-p)^k

Answer

P(X ≤ k) = 1 - (1-p)^k, representing the probability that the first success occurs by the k-th trial.

PMF - Probability Mass Function Practice Test