How is the variance of a negative binomial distribution expressed?

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How is the variance of a negative binomial distribution expressed?

The variance of a negative binomial distribution is given by the formula var(X) = rq/p^2, where r represents the number of successes, p is the probability of success on each trial, and q is the probability of failure, which can also be expressed as q = 1 - p. The rationale behind this formula is that the negative binomial distribution models the number of failures that occur before r successes in a series of independent Bernoulli trials. As the number of successes (r) increases, the variability in the number of failures (and therefore the overall count of trials until those successes occur) grows proportionally with the number of successes and inversely with the probability of success squared. In simple terms, as the probability of success decreases (making failures more likely), the variance increases more significantly due to the factor of p^2 in the denominator, which adjusts the variance according to how likely successes are in the trials. Thus, this variance representation captures the essence of the distribution's behavior appropriately.

The variance of a negative binomial distribution is given by the formula var(X) = rq/p^2, where r represents the number of successes, p is the probability of success on each trial, and q is the probability of failure, which can also be expressed as q = 1 - p.

The rationale behind this formula is that the negative binomial distribution models the number of failures that occur before r successes in a series of independent Bernoulli trials. As the number of successes (r) increases, the variability in the number of failures (and therefore the overall count of trials until those successes occur) grows proportionally with the number of successes and inversely with the probability of success squared.

In simple terms, as the probability of success decreases (making failures more likely), the variance increases more significantly due to the factor of p^2 in the denominator, which adjusts the variance according to how likely successes are in the trials. Thus, this variance representation captures the essence of the distribution's behavior appropriately.

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