# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is a important division of mathematics that takes up the study of random occurrence. One of the crucial ideas in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the number of trials required to obtain the initial success in a secession of Bernoulli trials. In this blog article, we will talk about the geometric distribution, derive its formula, discuss its mean, and offer examples.

## Definition of Geometric Distribution

The geometric distribution is a discrete probability distribution that describes the number of experiments required to achieve the initial success in a sequence of Bernoulli trials. A Bernoulli trial is a test which has two viable 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 tests are independent, which means that the consequence of one experiment does not affect the outcome of the next trial. 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 that depicts the amount of test required to achieve the initial success, k is the number of experiments required to achieve the first 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 likely value of the number 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 expected count of experiments needed to obtain the initial success. Such as if the probability of success is 0.5, then we anticipate to obtain the initial success after two trials on average.

## Examples of Geometric Distribution

Here are some essential examples of geometric distribution

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

Imagine we flip a fair coin until the first head turns up. The probability of success (getting a head) is 0.5, and the probability of failure (getting a tail) is also 0.5. Let X be the random variable which portrays the number of coin flips required to get the initial head. The PMF of X is given by:

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

For k = 1, the probability of obtaining 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 obtaining 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 until the initial six appears.

Suppose we roll a fair die till the first six appears. The probability of success (obtaining a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the irregular variable that portrays the count of die rolls needed to get the first 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 initial six on the initial 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 getting 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 concept in probability theory. It is applied to model a broad range of practical scenario, for instance the count of trials required to obtain the initial success in various situations.

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