Geometric distribution The geometric distribution models the time until the first success in a sequence of independent and identically distributed events. I...
Geometric distribution
The geometric distribution models the time until the first success in a sequence of independent and identically distributed events. In this context, a success is an event that occurs when a particular outcome in a sequence of independent events happens. The probability of a success on any given trial is constant and equal to p.
The probability mass function (PMF) for the geometric distribution is given by:
where:
X is the random variable representing the number of trials until the first success
k is the random variable representing the number of trials
p is the probability of a success on each trial
Negative binomial distribution
The negative binomial distribution models the number of successes before a specified number of failures in a sequence of independent and identically distributed events. A failure is an event that occurs when a particular outcome in a sequence of independent events does not happen. The probability of a success on any given trial is constant and equal to p.
The probability mass function (PMF) for the negative binomial distribution is given by:
where:
X is the random variable representing the number of successes before the number of failures
k is the random variable representing the number of successes
n is the random variable representing the number of trials
p is the probability of a success on each trial
Geometric and negative binomial distributions are used in a variety of applications, including queuing theory, probability theory, and statistics. They are used to model the time until a customer visits a store, the number of patients that visit a doctor on a given day, and the number of successes in a sequence of independent experiments