What is the expected value for a geometric distribution?

What is the expected value for a geometric distribution?

• A. E[X] = q/p
• B. E[X] = p/q
• C. E[X] = 1/p
• D. E[X] = (1 - q) / p

Answer: (C)

The expected value for a geometric distribution is derived from the nature of the distribution itself, which models the number of trials needed to get the first success in a series of independent Bernoulli trials (where each trial has only two possible outcomes, success or failure). In a geometric distribution, the probability of success on any given trial is denoted as \( p \), and the probability of failure is denoted as \( q = 1 - p \). The expected value, denoted as \( E[X] \), quantifies the average number of trials needed to achieve the first success. The formula for the expected value of a geometric distribution is given as \( E[X] = \frac{1}{p} \). This formula reflects the idea that as the probability of success \( p \) increases, the expected number of trials until the first success decreases, since a higher probability of success means you are likely to succeed sooner. In summary, this expected value is correct because it succinctly represents the average number of trials to achieve success in a scenario characterized by a constant probability of success \( p \) for each trial.

The expected value for a geometric distribution is derived from the nature of the distribution itself, which models the number of trials needed to get the first success in a series of independent Bernoulli trials (where each trial has only two possible outcomes, success or failure).

In a geometric distribution, the probability of success on any given trial is denoted as ( p ), and the probability of failure is denoted as ( q = 1 - p ). The expected value, denoted as ( E[X] ), quantifies the average number of trials needed to achieve the first success.

The formula for the expected value of a geometric distribution is given as ( E[X] = \frac{1}{p} ). This formula reflects the idea that as the probability of success ( p ) increases, the expected number of trials until the first success decreases, since a higher probability of success means you are likely to succeed sooner.

In summary, this expected value is correct because it succinctly represents the average number of trials to achieve success in a scenario characterized by a constant probability of success ( p ) for each trial.