Geometric Distribution:

Consider a sequence of Bernoulli trials (failure and success), the geometric distribution is used to find the number of failures before the first success. For a geometric distribution with probability of success, the probability that exactly x failures occur before the first success is

P(X=x) =

The geometric distribution is the only discrete distribution with the memory less property. The successive probabilities in this distribution form a geometric series, hence the name to the distribution.

• In baseball, a geometric distribution is useful in analyzing the probability of a batter earning a hit before three strikes, a success within 3 trials.
• In cost-benefit analyses, whether to fund research trials that, if successful, will earn the company some estimated profit, getting a success before the cost outweighs the potential gain.

The probability mass function, the probability that the x^th trial (out of x trials) is the first success is

f(x) = P(X= x) p   where 0 < p < 1, x = 1, 2, 3..

MEAN, μ = E(X) =

Variance,  = Var(X) = 

In Excel, we use the function NEGBINOMDIST(number_f, number_s, probability_s) where

Number_f  –  number of failures , Number_s   – the threshold number of successes(for geometric distribution,1) , Probability_s – the probability of a success on each trial.

Example:

A die is rolled until a 1 shows up. Using the function, resulting geometric distribution is shown graphically