# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is an essential department of mathematics which handles the study of random events. One of the crucial concepts in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the amount of tests required to obtain the initial success in a series of Bernoulli trials. In this blog, we will explain the geometric distribution, extract its formula, discuss its mean, and give examples.

## Meaning of Geometric Distribution

The geometric distribution is a discrete probability distribution that narrates the number of tests needed to accomplish the initial success in a succession of Bernoulli trials. A Bernoulli trial is a test that has two likely outcomes, usually indicated to as success and failure. Such as flipping a coin is a Bernoulli trial since it can likewise come up 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. Additionally, the chances of success remains constant throughout all the trials. We could indicate 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 depicts the amount of test required to attain the first success, k is the number of tests needed to attain the first 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 defined as the likely value of the amount of test 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 an individual Bernoulli trial.

The mean is the anticipated count of experiments needed to obtain the first success. Such as if the probability of success is 0.5, therefore we anticipate to attain the first success following two trials on average.

## Examples of Geometric Distribution

Here are some primary examples of geometric distribution

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

Suppose we flip an honest coin until the initial head turns up. The probability of success (obtaining 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 represents the count of coin flips needed to achieve 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 achieving the first head on the first flip is:

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

For k = 2, the probability of obtaining 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 first head on the third flip is:

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

And so on.

Example 2: Rolling a fair die up until the first six appears.

Let’s assume we roll an honest die up until the initial six shows up. The probability of success (getting a six) is 1/6, and the probability of failure (getting any other number) is 5/6. Let X be the random variable which represents the number of die rolls needed to achieve 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 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 getting 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 forth.

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The geometric distribution is an essential theory in probability theory. It is utilized to model a wide range of real-life phenomena, for instance the count of tests required to obtain the first success in different situations.

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