# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is an essential department of mathematics which handles the study of random occurrence. One of the important ideas in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the amount of tests required to get the first success in a sequence of Bernoulli trials. In this blog article, we will talk about the geometric distribution, extract its formula, discuss its mean, and give examples.

## Definition of Geometric Distribution

The geometric distribution is a discrete probability distribution which describes the amount of experiments required to accomplish the initial success in a succession of Bernoulli trials. A Bernoulli trial is a trial that has two possible outcomes, usually referred to as success and failure. Such as tossing a coin is a Bernoulli trial because it can either turn out to be heads (success) or tails (failure).

The geometric distribution is utilized when the experiments are independent, which means that the consequence of one trial doesn’t impact the outcome of the upcoming test. In addition, the chances of success remains unchanged across 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 provided by the formula:

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

Where X is the random variable which represents the number of test required to get the initial success, k is the count of experiments needed to obtain the initial success, p is the probability of success in an individual Bernoulli trial, and 1-p is the probability of failure.

Mean of Geometric Distribution:

The mean of the geometric distribution is explained as the likely value of the number of experiments required to achieve the initial success. The mean is stated in the formula:

μ = 1/p

Where μ is the mean and p is the probability of success in an individual Bernoulli trial.

The mean is the expected number of tests needed to obtain the initial success. For example, if the probability of success is 0.5, then we expect to attain the initial success after two trials on average.

## Examples of Geometric Distribution

Here are few primary examples of geometric distribution

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

Suppose we flip a fair coin till the first head appears. The probability of success (obtaining a head) is 0.5, and the probability of failure (getting a tail) is as well as 0.5. Let X be the random variable that depicts the number of coin flips needed to obtain the first 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 obtaining 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 getting the initial head on the second flip is:

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

For k = 3, the probability of achieving 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 up until the first six shows up.

Suppose we roll a fair die until the initial six appears. The probability of success (getting a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the random variable that portrays the number of die rolls needed to get the initial six. The PMF of X is stated as:

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

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

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

For k = 2, the probability of obtaining 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 initial six on the third roll is:

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

And so forth.

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The geometric distribution is a crucial concept in probability theory. It is applied to model a wide range of real-world phenomena, for instance the number of tests required to obtain the initial success in several situations.

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