2.7.1 Geometric

Question 1

What is a geometric distribution?

Answer 1

A discrete probability distribution that models the number of independent Bernoulli trials needed to get the first success.

Question 2

What are the key assumptions of a geometric distribution?

Answer 2

1. Each trial has only two outcomes (success or failure).
2. The probability of success stays the same for every trial.
3. The trials are independent.
4. The process continues until the first success occurs.

Question 3

How do we denote that a random variable ( X ) follows a geometric distribution?

Answer 3

( X sim ext{Geometric}(p) ), where ( p ) is the probability of success on each trial.

Question 4

What is the probability that the first success happens on the ( k )-th trial?

Answer 4

P(X = k) = (1 - p)^{k - 1} p
This means that the first ( k - 1 ) trials must all be failures, followed by a success.

Question 5

How does the geometric distribution differ from the binomial distribution?

Answer 5

• The binomial distribution models the number of successes in a fixed number of trials.
• The geometric distribution models the number of trials until the first success.

Question 6

What is the geometric survival function which indicates the probability that the first success happens after ( k ) trials?

Answer 6

P(X > k) = (1 - p)^k

Question 7

What is the expected value of a geometric r.v.?

Answer 7

E[X] =1/p

Question 8

What is the variance of a geometric distribution?

Answer 8

(1-p)/p^2

Question 9

What does the memoryless property of the geometric distribution state?

Answer 9

Question 10

How do the PMF, mean, and variance change in the failure defintion of a geometric r.v.?

Answer 10

since y = x - 1 or failures= successes - 1