Geometric Distribution

	Geometric Distribution

The geometric distribution is similar to the binomial distribution in that the probability of occurrence is constant from trial to trial and the trials are independent. The binomial distribution models situations where the number of trials is fixed, and the random variable is the number of successes. The geometric distribution requires exactly 1 success, and the random variable is the number of trials required to obtain the first success. The geometric distribution is a special case of the negative binomial distribution. The negative binomial distribution models the number of trials required to obtain m successes, and m is not required to be equal to one.

The geometric probability density function is

where p(x,p) is the probability that the first success occurs on the xth trial given a probability of success on a single trial of p.

The probability that more than n trials is required to obtain the first success is

The mean and variance of the geometric distribution are

Example
The probability of an enemy aircraft penetrating friendly airspace is 0.01. What is the probability that the first penetration of friendly airspace is accomplished by the 80th aircraft to attempt the penetration of friendly airspace?

Solution
The probability that the first success occurs on the 80th trial with a probability of success of 0.01 on each trial is

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Example
The probability of an enemy aircraft penetrating friendly airspace is 0.01. What is the probability that it will take more than 80 attempts to penetrate friendly airspace?

Solution
The probability that it will take more than 80 attempts to penetrate friendly airspace with a probability of success of 0.01 on each trial is

Click Here to download this solution in Microsoft Excel.