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.
Definition of Geometric distribution
A discrete random variable X is said to have a geometric distribution if its probability density function is defined as,
![\[ f(x;p)=pq^{x}; x=0,1,2,...,\infty \]](https://www.statisticalaid.com/wp-content/ql-cache/quicklatex.com-7adb069857b7810d108a2a19491fccdf_l3.png "Rendered by QuickLaTeX.com")
where p is the only parameter of of geometric distribution which satisfy 0<=p<=1 and p+q=1.
Assumptions
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.
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.
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.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.