# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is a important branch of math which deals with the study of random occurrence. One of the essential theories in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the amount of experiments required to get the initial success in a sequence of Bernoulli trials. In this article, we will define the geometric distribution, extract its formula, discuss its mean, and offer examples.

## Explanation 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 an experiment that has two possible outcomes, usually referred to as success and failure. Such as tossing a coin is a Bernoulli trial since it can either come up heads (success) or tails (failure).

The geometric distribution is used when the experiments are independent, which means that the outcome of one trial does not affect the outcome of the upcoming trial. Furthermore, the chances of success remains unchanged across all the trials. We could 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 that portrays the amount of trials needed to get the initial success, k is the number of tests required to achieve 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 expected value of the number of trials required to obtain 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 expected number of trials required to get the initial success. For example, if the probability of success is 0.5, then we anticipate to attain the first success after two trials on average.

## Examples of Geometric Distribution

Here are few primary examples of geometric distribution

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

Imagine we flip a fair coin until the initial head shows up. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is also 0.5. Let X be the random variable which depicts the number of coin flips needed to obtain the initial head. The PMF of X is provided 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 on.

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

Let’s assume we roll an honest die up until the first six turns up. The probability of success (achieving a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the random variable which 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 obtaining 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 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 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 an essential theory in probability theory. It is applied to model a wide range of real-world phenomena, for example the count of tests required to obtain the first success in several scenarios.

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