# Geometric Distribution Formula

## Geometric Distribution Formula

In a Bernoulli trial, a Geometric Distribution Formula—a kind of discrete probability distribution—represents the probability of the number of failures that must occur before success is attained. A test called a Bernoulli trial can only result in success or failure. To put it another way, a Bernoulli trial is carried out repeatedly until success is attained, at which point it is stopped.

The Geometric Distribution Formula is frequently employed in a variety of real-world situations. For instance, in the financial sector, a cost-benefit analysis may use the Geometric Distribution Formula to assess the financial benefits of a certain decision.

A flat shape’s area, a solid object’s volume, and the calculation of length or distance are all determined by measurement in Geometry. Geometric figures’ perimeters, areas, capacities, surface areas, and volumes are calculated using measurements. The terms perimeter (the space around plane shapes), area (the space occupied by the shape), volume (the amount of space occupied by a solid), and surface area (a solid’s total surface area) all refer to different aspects of solids.

## What is Geometric Distribution?

One of the probability distributions that plays a part in forecasting the outcomes of continuous occurrences is the Geometric Distribution Formula. The geometric distribution calculates the average time between successes for a set of Bernoulli trials.

The Geometric Distribution Formula is one of the most significant discrete distributions since it is frequently utilised to solve probability calculations. X is the number of Bernoulli trials needed to get one success in a geometric distribution.

A Geometric Distribution Formula is a discrete probability distribution of a random variable “x,” and it meets the conditions listed below:

• A phenomenon that has undergone several tests,
• There are only two possible results for each trial: success or failure. and
• For each trial, the likelihood of success is the same.

## Geometric Distribution Definition

The Geometric Distribution Formula is a discrete probability distribution that depicts the likelihood of experiencing success for the first time following a string of failures. A Geometric Distribution Formula may undergo an infinite number of trials before experiencing its first success.

## Geometric Distribution Formula

The probability distribution, known as the Geometric Distribution Formula, determines the likelihood of the first success after k tries. Using the geometric method, one may also calculate the likelihood of achieving x success in k trials.

The cumulative distribution function (CDF) and the probability mass function (PMF) can be used to calculate a geometric distribution. Here, p represents the likelihood that a trial would be successful and q for its failure.

The likelihood of the number of subsequent failures in a Bernoulli trial before success is attained is represented by the geometric distribution, which is a discrete probability distribution. An experiment called a Bernoulli trial has only two possible results: success or failure. To put it another way, a Bernoulli trial is iterated in a geometric distribution until success is attained, at which point it is halted.

In numerous situations in real life, the Geometric Distribution Formula is frequently applied. For instance, the geometric distribution is employed in the financial sector to assess the financial rewards of a particular course of action.

### Geometric Distribution PMF

It is possible to define the probability mass function as the likelihood that a discrete random variable, X, will exactly equal a specific value, x. The following is the formula for the geometric distribution PMF:

P(X = x) = (1 – p)x – 1p

where 0 < p ≤ 1.

### Geometric Distribution CDF

The probability that a random variable, X, will have a value that is less than or equal to x can be defined as the cumulative distribution function of a random variable, X, that is assessed at a point, x. The distribution function is an alternative name for it. The following is the Geometric Distribution Formula CDF:

P(X ≤ x) = 1 – (1 – p)x

### Mean of Geometric Distribution

The anticipated value of the Geometric Distribution Formula is equal to the geometric distribution’s mean. The weighted average of all X values can be used to define the expected value of a random variable, X. The following is the Geometric Distribution Formula mean:

E[X] = 1 / p

### Variance of Geometric Distribution

Variance is a measure of dispersion that evaluates how far the data in a distribution are dispersed from the mean. The following is the Geometric Distribution Formula variance:

Var[X] = (1 – p) / p2

### Standard Deviation of Geometric Distribution

The square root of the variance can be used to define the standard deviation. The distribution’s divergence from the mean is also indicated by the standard deviation. The following is the formula for the Geometric Distribution Formula standard deviation:

S.D. = √VAR[X]

S.D. = √1-p / p

### Binomial Vs Geometric Distribution

There are two possible outcomes for a trial in both geometric and binomial distribution: success or failure. Additionally, each experiment will have the same chance of success.

The only success that matters in a geometric distribution is the first one. The random variable, X, keeps track of how many attempts are necessary to achieve the first success. A random variable, X, counts how many of the trials in a binomial distribution are successful. There are a fixed number of trials in the distribution.

### Examples on Geometric Distribution

• Example 1: Suppose one is playing a game of darts. The probability of success is 0.4. What is the probability that they will hit the bullseye on the third try?

Solution: As students are looking for the first success, thus, Geometric Distribution Formula has to be used.

p = 0.4

P(X = x) = (1 – p)x – 1p

P(X = 3) = (1 – 0.4)3 – 1(0.4)

P(X = 3) = (0.6)2(0.4) = 0.144

Answer: The probability that one will hit the bullseye on the third try is 0.144

• Example 2: A manufacturing factory finds 3 in every 60 light bulbs defective. What is the probability that the first defective light bulb will be found when the 6th one is tested?

Solution: As the probability of the first defective light bulb needs to be determined hence, this is a geometric distribution.

p = 3 / 60 = 0.05

P(X = x) = (1 – p)x – 1p

P(X = 6) = (1 – 0.05)6 – 1(0.05)

P(X = 6) = (0.95)5(0.05)

P(X = 6) = 0.0386

Answer: The probability that the first defective light bulb is found on the 6th trial is 0.0368

### Practice Questions on Geometric Distribution

• Problem 1: Calculate the probability density of the Geometric Distribution Formula if the value of p is 0.42; x = 1,2,3,…….., and also find out the mean and variance.

Solution:

Given that, p = 0.42 and the value of x is 1,2,3,……………

The probability density of the Geometric Distribution Formula function,

P(x) = p

(1−p)x−1

; x = 1,2,3,…

P(x) = 0; otherwise

(x) = 0.42

(1−0.42)x−1

P(x) = 0 otherwise

Mean = 1p

=10.42

= 2.380

Variance = 1−pp2

Variance = 1−0.420.422

Variance = 3.288

• Problem 2: Find the probability density of geometric distribution if the value of p is 0.42; x = 1,2,3 and also calculate the mean and variance.

Solution:

Given that p = 0.42 and the value of x = 1, 2, 3

The probability density of the Geometric Distribution Formula is

P(x) = p (1-p) x-1; x =1, 2, 3

P(x) = 0; otherwise

P(x) = 0.42 (1- 0.42)

P(x) = 0; Otherwise

Mean= 1/p = 1/0.42 = 2.380

Variance = 1-p/ p2

= 1-0.42 /(0.42)2

= 3.287

## 1. How Should a Geometric Distribution Be Written?

A Geometric Distribution Formula is represented by the notation XG(p)XG (p). Here, the random variable is denoted by X, G denotes its geometric distribution, and the chance of each trial’s success is denoted by p.

## 2. What does the statistical Geometric Distribution Formula mean?

A geometric distribution is a type of probability distribution that indicates how many times a Bernoulli trial must be carried out before the first success is obtained following a string of failures.

## 3. What distinguishes the geometric distribution from the binomial distribution?

The number of trials in a binomial distribution is fixed, in contrast to a geometric distribution with an infinite number of trials—the number of trials that the random variable determines success. However, with a geometric distribution, the random variable keeps track of how many tries are necessary to achieve the first success.

## 4. What Purposes Does Geometric Distribution Serve?

Geometric distribution applications are widely used in various fields, including Manufacturing, Computer Science, Sports, and Finance. It determines the probability of success after a specific number of trials.

## 5. What are the statistical characteristics of geometric distribution?

The presence of many values reveals details about a particular probability distribution. The average one may anticipate from numerous trials is given by the mean or anticipated value of the distribution. The value with the highest likelihood of appearing is the distribution mode. When a distribution has an outlier (particularly huge or small values), which confounds the mean, the median is a useful alternative measure of central tendency. The way that data is “spread out” depends on the distribution variance. While the median is not typically computed, the three numbers mean, mode and variance are typically utilised for geometric distribution.