Geometric Distribution: Definition, Properties and Applications

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A geometric distribution is a discrete probability distribution that illustrates the probability that a Bernoulli trial will result in multiple failures before success. A Bernoulli trial is an experiment that can have only two possible outcomes, i.e., success or failure. In a geometric distribution, a Bernoulli trial is essentially repeated until success is attained.

It is widely used in several real-life activities. For example, in the financial industry, it is used to do a cost-benefit analysis to estimate the financial benefits of making a certain decision.

Geometric Distribution

Definition of Geometric distribution

A discrete random variable X is said to have a geometric distribution if its probability density function is defined as,

    \[ f(x;p)=pq^{x}; x=0,1,2,...,\infty  \]

 where p is the only parameter of of geometric distribution which satisfy  0<=p<=1 and p+q=1.


There are three main assumptions.

  • The trials must be independent. 
  • Each experiment can only result in one of two outcomes: success or failure. 
  • For each trial, the probability of success is denoted by p.
Geometric distribution curve

Properties of Geometric Distribution

  • Geometric distribution follows the lack memory property.
  • The mean of geometric distribution is .
  • The variance of geometric distribution is .
  • Moment generating function of geometric distribution is .
  • The mean of geometric distribution is smaller then its variance, since q/p2 > q/p.

Application of Geometric distribution

  • The Geometric distribution is used in Markov chain models.
  • It is used in meteorological modes of weather cycles and precipitation amounts.
  • The geometric distribution is often referred to as the failure time distribution.
  • It would be used to describe the number of interviews that have to be conducted by a selection board to appoint the first acceptable candidate.

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