# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is a important branch of math that takes up the study of random events. One of the crucial ideas in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the amount of trials needed to get the first success in a series of Bernoulli trials. In this 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 tests required to accomplish the initial success in a sequence of Bernoulli trials. A Bernoulli trial is a trial which has two possible results, usually indicated 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 applied when the experiments are independent, meaning that the result of one test does not affect the outcome of the next test. Furthermore, the chances of success remains same throughout all the tests. We could signify 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 that depicts the number of trials needed to attain the first success, k is the count 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 explained as the expected value of the number of trials required to achieve the first success. The mean is stated in 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 count of experiments needed to obtain the initial success. For example, if the probability of success is 0.5, therefore we anticipate to attain the initial success following two trials on average.

## Examples of Geometric Distribution

Here are some primary examples of geometric distribution

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

Let’s assume we flip a fair coin till the initial 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 represents the count 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 first 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 obtaining 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 until the first six turns up.

Let’s assume we roll an honest die up until the first six turns up. The probability of success (obtaining a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the random variable which portrays the count of die rolls needed to get the initial six. The PMF of X is provided 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 initial roll is:

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

For k = 2, the probability of getting the initial 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 theory in probability theory. It is utilized to model a broad array of real-life scenario, such as the count of trials needed to obtain the initial success in various situations.

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