April 13, 2023

Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is ac crucial division of math that handles the study of random occurrence. One of the essential ideas in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the number of tests required to obtain the first success in a secession of Bernoulli trials. In this article, we will explain the geometric distribution, derive its formula, discuss its mean, and provide examples.

Definition of Geometric Distribution

The geometric distribution is a discrete probability distribution which portrays the amount of experiments needed to achieve the initial success in a series of Bernoulli trials. A Bernoulli trial is a trial which has two likely results, usually referred to as success and failure. For example, flipping a coin is a Bernoulli trial since it can either come up heads (success) or tails (failure).


The geometric distribution is utilized when the tests are independent, which means that the consequence of one experiment does not affect the outcome of the upcoming test. Additionally, the chances of success remains unchanged throughout all the trials. We can denote the probability of success as p, where 0 < p < 1. The probability of failure is then 1-p.

Formula for Geometric Distribution

The probability mass function (PMF) of the geometric distribution is specified by the formula:


P(X = k) = (1 - p)^(k-1) * p


Where X is the random variable which represents the number of test needed to get the first success, k is the number of experiments required to achieve the initial success, p is the probability of success in a single Bernoulli trial, and 1-p is the probability of failure.


Mean of Geometric Distribution:


The mean of the geometric distribution is described as the expected value of the amount of trials required to achieve the first success. The mean is given by the formula:


μ = 1/p


Where μ is the mean and p is the probability of success in a single Bernoulli trial.


The mean is the anticipated number of experiments required to obtain the first success. Such as if the probability of success is 0.5, therefore we anticipate to obtain the initial success after two trials on average.

Examples of Geometric Distribution

Here are handful of basic examples of geometric distribution


Example 1: Flipping a fair coin up until the first head turn up.


Suppose we toss an honest coin till the initial head appears. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is as well as 0.5. Let X be the random variable that portrays the count of coin flips needed to obtain the initial head. The PMF of X is stated as:


P(X = k) = (1 - 0.5)^(k-1) * 0.5 = 0.5^(k-1) * 0.5


For k = 1, the probability of getting the initial head on the first flip is:


P(X = 1) = 0.5^(1-1) * 0.5 = 0.5


For k = 2, the probability of achieving the first head on the second flip is:


P(X = 2) = 0.5^(2-1) * 0.5 = 0.25


For k = 3, the probability of getting the initial head on the third flip is:


P(X = 3) = 0.5^(3-1) * 0.5 = 0.125


And so forth.


Example 2: Rolling an honest die till the initial six appears.


Let’s assume we roll a fair die till the initial six appears. The probability of success (getting a six) is 1/6, and the probability of failure (obtaining any other number) is 5/6. Let X be the random variable which represents the count of die rolls required to obtain the initial six. The PMF of X is given by:


P(X = k) = (1 - 1/6)^(k-1) * 1/6 = (5/6)^(k-1) * 1/6


For k = 1, the probability of achieving the first six on the first roll is:


P(X = 1) = (5/6)^(1-1) * 1/6 = 1/6


For k = 2, the probability of achieving the first six on the second roll is:


P(X = 2) = (5/6)^(2-1) * 1/6 = (5/6) * 1/6


For k = 3, the probability of achieving the first six on the third roll is:


P(X = 3) = (5/6)^(3-1) * 1/6 = (5/6)^2 * 1/6


And so on.

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The geometric distribution is an essential theory in probability theory. It is applied to model a broad range of practical scenario, for instance the number of trials needed to get the first success in different situations.


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