What is the formula for the variance of a geometric distribution?

What is the formula for the variance of a geometric distribution?

• A. q/p
• B. q/p^2
• C. pq
• D. (1-q)/p

Answer: (B)

The variance of a geometric distribution, which models the number of trials until the first success in a series of independent Bernoulli trials, can be determined using the parameters of the distribution. In this context, let p represent the probability of success on each trial, while q represents the probability of failure (where q = 1 - p). The variance for a geometric distribution is derived from the properties of expected value and the concept of variance. Specifically, the formula for the variance is given by: \[ \text{Variance} = \frac{q}{p^2} \] This means that the variance increases as the probability of failure (q) increases, and it is inversely related to the square of the probability of success (p). This is essential because it highlights how variability in the number of trials until the first success can depend heavily on the likelihood of success occurring in each trial. Because the geometric distribution is concentrated on the tail end of the distribution (i.e., it counts the number of trials until the first success), it is expected that the variance will reflect this tendency, leading to the formula involving q and the square of p. Thus, the choice that correctly represents the variance of a geometric distribution is based on this

The variance of a geometric distribution, which models the number of trials until the first success in a series of independent Bernoulli trials, can be determined using the parameters of the distribution. In this context, let p represent the probability of success on each trial, while q represents the probability of failure (where q = 1 - p).

The variance for a geometric distribution is derived from the properties of expected value and the concept of variance. Specifically, the formula for the variance is given by:

[

\text{Variance} = \frac{q}{p^2}

]

This means that the variance increases as the probability of failure (q) increases, and it is inversely related to the square of the probability of success (p). This is essential because it highlights how variability in the number of trials until the first success can depend heavily on the likelihood of success occurring in each trial.

Because the geometric distribution is concentrated on the tail end of the distribution (i.e., it counts the number of trials until the first success), it is expected that the variance will reflect this tendency, leading to the formula involving q and the square of p.

Thus, the choice that correctly represents the variance of a geometric distribution is based on this