Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential department of mathematics that takes up the study of random events. One of the crucial theories in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the number of experiments required to obtain the initial success in a series of Bernoulli trials. In this article, we will explain the geometric distribution, extract its formula, discuss its mean, and give examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution which narrates the number of tests required to reach the initial success in a sequence of Bernoulli trials. A Bernoulli trial is a trial that has two likely outcomes, usually referred to as success and failure. For example, flipping a coin is a Bernoulli trial since it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is used when the experiments are independent, which means that the consequence of one experiment doesn’t impact the outcome of the next trial. Furthermore, the chances of success remains same throughout all the tests. We can 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 provided by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which depicts the number of trials required to achieve the initial success, k is the count of experiments needed to obtain the first 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 defined as the expected value of the number of trials needed to get 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 an individual Bernoulli trial.
The mean is the anticipated count of experiments required to obtain the first success. For example, if the probability of success is 0.5, therefore we expect to get the initial success following two trials on average.
Examples of Geometric Distribution
Here are few essential examples of geometric distribution
Example 1: Tossing a fair coin until the first head appears.
Imagine we flip a fair coin until the first head turns up. The probability of success (obtaining a head) is 0.5, and the probability of failure (getting a tail) is as well as 0.5. Let X be the random variable that portrays the number of coin flips required 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 getting the first 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 until the initial six appears.
Suppose we roll an honest die till the initial six turns up. The probability of success (achieving a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the irregular variable which represents the number of die rolls required to achieve 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 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 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 on.
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The geometric distribution is a crucial theory in probability theory. It is applied to model a broad range of practical scenario, such as the number of trials required to get the first success in several scenarios.
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