Geometric Distribution - Definition, Formula, Mean, Examples

Geometric Distribution

Geometric distribution is a type of discrete probability distribution that represents the probability of the number of successive failures before a success is obtained in a Bernoulli trial. A Bernoulli trial is an experiment that can have only two possible outcomes, ie., success or failure. In other words, in a geometric distribution, a Bernoulli trial is repeated until success is obtained and then stopped.

The geometric probability distribution is widely used in several real-life scenarios. For example, in financial industries, geometric distribution is used to do a cost-benefit analysis to estimate the financial benefits of making a certain decision. In this article, we will study the meaning of geometric distribution, examples, and certain related important aspects.

1.	What is Geometric Distribution?
2.	Geometric Distribution Formula
3.	Geometric Distribution Mean (Expected Value)
4.	Variance of Geometric Distribution
5.	Standard Deviation of Geometric Distribution
6.	Binomial Vs Geometric Distribution
7.	FAQs on Geometric Distribution

What is Geometric Distribution?

The geometric distribution is a probability distribution that models the number of trials required to achieve the first success in a sequence of independent Bernoulli trials, where each trial has a constant probability of success.

i.e., Geometric distribution that is based on three important assumptions. These are listed as follows.

• The trials being conducted are independent.
• There can only be two outcomes of each trial - success or failure.
• The success probability, denoted by p, is the same for each trial.

Geometric Distribution Definition

Geometric distribution can be defined as a discrete probability distribution that represents the probability of getting the first success after having a consecutive number of failures. A geometric distribution can have an indefinite number of trials until the first success is obtained.

Geometric Distribution Example

Suppose a dice is repeatedly rolled until "3" is obtained. We know that the probability of getting "3" is p = 1 / 6 and let the random variable X take the values 1, 2, and 3.

• The probability of rolling a 3 in the first trial is 1/6.
• The probability of rolling a 3 in the second trial for the first time is 5/6 × 1/6 = 5/36. Here, 5/6 is the prob of rolling a number that is NOT 3 in the first trial.
• Similarly, the probability of rolling a 3 in the third trial for the first time is, (5/6)2 × 1/6 = 25/216.

Geometric Distribution Formula

There are two geometric probability formulas:

• Geometric distribution PMF: P(X = x) = (1 - p)x - 1p
• Geometric distribution CDF: P(X ≤ x) = 1 - (1 - p)x

A geometric distribution can be described by both the probability mass function (PMF) and the cumulative distribution function (CDF). The probability of success of a trial is denoted by p and failure is given by q. Here, q = 1 - p. A discrete random variable, X, that has a geometric probability distribution is represented as \(X\sim G(p)\). Given below are the formulas for the PMF and CDF of a geometric distribution.

Geometric Distribution PMF

The probability mass function can be defined as the probability that a discrete random variable, X, will be exactly equal to some value, x. The formula for geometric distribution PMF is given as follows:

P(X = x) = (1 - p)x - 1p

where, 0 < p ≤ 1.

Geometric Distribution CDF

The cumulative distribution function of a random variable, X, that is evaluated at a point, x, can be defined as the probability that X will take a value that is lesser than or equal to x. It is also known as the distribution function. The formula for the geometric distribution CDF is given as follows:

P(X ≤ x) = 1 - (1 - p)x

Geometric Distribution Mean (Expected Value)

The mean of geometric distribution is also the expected value of the geometric distribution. The expected value of a random variable, X, can be defined as the weighted average of all values of X. The formula for the mean of a geometric distribution is given as follows:

E[X] = 1 / p

Variance of Geometric Distribution

Variance can be defined as a measure of dispersion that checks how far the data in a distribution is spread out with respect to the mean. The formula for the variance of a geometric distribution is given as follows:

Var[X] = (1 - p) / p2

Standard Deviation of Geometric Distribution

The standard deviation can be defined as the square root of the variance. The standard deviation also gives the deviation of the distribution with respect to the mean. The formula for the standard deviation of a geometric distribution is as follows:

S.D. = \(\sqrt{Var[X]}\)

S.D. = \(\frac{\sqrt{1 - p}}{p}\)

Binomial Vs Geometric Distribution

In both geometric distribution and binomial distribution, there can be only two outcomes of a trial, either success or failure. Furthermore, the probability of success will be the same for each trial. The difference between binomial distribution and geometric distribution is given in the table below.

Geometric Distribution	Binomial Distribution
A geometric distribution is concerned with the first success only. The random variable, X, counts the number of trials required to obtain that first success.	In a binomial distribution, there are a fixed number of trials and the random variable, X, counts the number of successes in those trials.
The probability mass function is given by PMF = (1 - p)x - 1p	The probability mass function is given by PMF = \(\binom{n}{x}p^{x}(1 - p)^{n-x}\)
Mean = 1 / p, Variance = (1 - p) / p2	Mean = np, Variance = np(1-p)

Important Notes on Geometric Distribution

• The geometric distribution is a discrete probability distribution where the random variable indicates the number of Bernoulli trials required to get the first success.
• The probability mass function of a geometric probability distribution is (1 - p)x - 1p and the cumulative distribution function is 1 - (1 - p)x.
• The mean of a geometric distribution is 1 / p and the variance is (1 - p) / p2.

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