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# Geometric Distribution

Probability is the chances of occurrence of an event. Probability deals with two types of distributions – geometric and binomial. Before coming to geometric distribution, let us know what the Bernoulli trial is.

Bernoulli trial is a way to find the probability of an event. It has only two outcomes – success and failure. The geometric distribution is a way to find the probability of these events if they occur repeatedly.

In simpler terms, the geometric distribution represents the probability of an event when success is obtained in a Bernoulli trial. The trials are repeated infinitely till success comes, even though failure occurs repeatedly. The trial ends once a success occurs.

For example, if we need to find the probability of 3 in a dice occurrence, we have to roll the dice till 3 comes. The trial will end if 3 comes in one trial or after 200 trials.

The geometric distribution is mostly used in financial sectors to do cost-benefit analyses, estimate cost and income benefits, and make a certain financial decision based on the studies and calculations after applying geometric distributions.

There are three assumptions that need to keep in mind while dealing with geometric distributions:
1) The trials conducted in this distribution are independent.
2) There can only be two outcomes of every event – success and failure.
3) The probability of success is denoted by p, and a failure by q, in most of the texts.

## Formula for geometric distribution

P (X = x) = (1 – p)⁻¹ p

P (X ≤ x) = 1 – (1 – p)ˣ

Where, p = probability of success

X = discrete random variable

x = a value where analysis is to done

The first formula is derived from the probability mass function (PMF). And the second formula is derived from the cumulative distribution function (CDF).

## Mean of geometric distribution

The mean value of a geometric distribution is known as the expected value of the geometric distribution. The expected value is represented by X, which is a weighted average of all values of X. The mean is denoted by:

E [X] = 1/p

## Variance of geometric distribution

The variance in a geometric distribution checks how far the data is spread out with respect to the mean within the distribution. The formula to derive a variance is:

Var [X] = (1 – p) / p²

## Standard deviation of geometric distribution

The root of variance is known as the standard deviation. The formula of standard deviation is:

## Difference between geometric and binomial distributions

 Geometric Distribution Binomial Distribution A geometric distribution deals with the first success only. The random variable, X, denotes the number of trials that occur to obtain that first success. In a binomial distribution, there are a fixed number of trials. The random variable, X, counts the number of successes that occur in those trials. The probability mass function is given by PMF = (1 - p)ˣ⁻¹ p The probability mass function is given by PMF = pˣ (1 − p)ⁿ⁻ˣ Mean = 1 / p, Variance = (1 - p) / p² Mean = np, Variance = np (1 - p)

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