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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 reallife scenarios. For example, in financial industries, geometric distribution is used to do a costbenefit 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.
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  1}p
 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  1}p
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) / p^{2}
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  1}p 
The probability mass function is given by PMF = \(\binom{n}{x}p^{x}(1  p)^{nx}\) 
Mean = 1 / p, Variance = (1  p) / p^{2}  Mean = np, Variance = np(1p) 
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  1}p 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) / p^{2}.
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Geometric Distribution Examples

Example 1: If a patient is waiting for a suitable blood donor and the probability that the selected donor will be a match is 0.2, then find the expected number of donors who will be tested till a match is found including the matched donor.
Solution: As we are looking for only one success this is a geometric distribution.
p = 0.2
E[X] = 1 / p = 1 / 0.2 = 5
Answer: The expected number of donors who will be tested till a match is found is 5 (including the donor).

Example 2: Suppose you are playing a game of darts. The probability of success is 0.4. What is the probability that you will hit the bullseye on the third try?
Solution: As we are looking for the first success, thus, geometric probability distribution has to be used.
p = 0.4
P(X = x) = (1  p)^{x  1}p
P(X = 3) = (1  0.4)^{3}^{  1}(0.4)
P(X = 3) = (0.6)^{2}(0.4) = 0.144
Answer: The probability that you will hit the bullseye on the third try is 0.144

Example 3: A light bulb manufacturing factory finds 3 in every 60 light bulbs defective. What is the probability that the first defective light bulb with be found when the 6^{th} one is tested?
Solution: As the probability of the first defective light bulb needs to be determined hence, this is a geometric distribution.
p = 3 / 60 = 0.05
P(X = x) = (1  p)^{x  1}p
P(X = 6) = (1  0.05)^{6}^{  1}(0.05)
P(X = 6) = (0.95)^{5}(0.05)
P(X = 6) ≈ 0.0387
Answer: The probability that the first defective light bulb is found on the 6th trial is 0.0367
FAQs on Geometric Distribution
What is Geometric Distribution in Statistics?
Geometric distribution is a probability distribution that describes the number of times a Bernoulli trial needs to be conducted in order to get the first success after a consecutive number of failures.
What are the Geometric Distribution Formulas?
The probability mass function and the cumulative distribution function formulas of a geometric distribution are given below:
 PMF: P(X = x) = (1  p)^{x  1}p
 CDF: P(X ≤ x) = 1  (1  p)^{x}
In addition, the following are the geometric probability formulas for mean, variance, and standard deviation.
 Mean (or) Expected value = 1/p
 Variance = (1  p) / p^{2}
 Standard deviation = \(\frac{\sqrt{1  p}}{p}\)
What is the Expected Value of a Geometric Distribution?
The expected value of a Geometric Distribution is given by E[X] = 1 / p. The expected value is also the mean of the geometric distribution.
How Do You Write a Geometric Distribution?
The notation of a geometric distribution is given by \(X\sim G(p)\). Here, X is the random variable, G indicates that the random variable follows a geometric distribution and p is the probability of success for each trial.
What is Geometric Probability Used For?
The applications of geometric distribution see widespread use in several industries such as finance, sports, computer science, and manufacturing companies. It is used to find the likelihood of a success when given a certain number of trials.
What is the Standard Deviation of a Geometric Distribution?
The standard deviation is the square root of the variance. It helps to measure the dispersion of the distribution about the mean of the given data. The standard deviation of a geometric probability is given as \(\frac{\sqrt{1  p}}{p}\).
What is the Difference Between Binomial Distribution and Geometric Distribution?
The main difference between a binomial distribution and a geometric distribution is that the number of trials in a binomial distribution is fixed. The random variable shows the number of successes in those trials. However, in a geometric distribution, the random variable counts the number of trials that will be required in order to get the first success.
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