# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is a important department of math which deals with 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 obtain the initial success in a series of Bernoulli trials. In this blog, we will talk about the geometric distribution, derive its formula, discuss its mean, and give examples.

## Explanation of Geometric Distribution

The geometric distribution is a discrete probability distribution that narrates the number of trials required to achieve the first success in a series of Bernoulli trials. A Bernoulli trial is a test that has two possible outcomes, usually referred to as success and failure. For instance, 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 trials are independent, which means that the outcome of one trial doesn’t impact the outcome of the next test. Furthermore, the probability of success remains constant across all the trials. We can 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 represents the number of trials needed to get the initial success, k is the number of experiments needed to attain 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 anticipated value of the number of experiments needed to achieve the initial 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 anticipated number of trials required to get the initial success. For instance, if the probability of success is 0.5, then we anticipate to obtain the first success following two trials on average.

## Examples of Geometric Distribution

Here are handful of primary examples of geometric distribution

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

Suppose we toss a fair coin until the first head turns 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 represents 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 first 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 on.

Example 2: Rolling an honest die up until the first six shows up.

Suppose we roll an honest die until the first six turns up. The probability of success (achieving a six) is 1/6, and the probability of failure (achieving all other number) is 5/6. Let X be the irregular variable which depicts the number of die rolls needed to obtain the first six. The PMF of X is given by:

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 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 obtaining 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 a crucial theory in probability theory. It is utilized to model a wide array of real-life phenomena, such as the count of trials required to achieve the initial success in various scenarios.

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