Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential branch of math that takes up the study of random events. One of the important concepts in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the amount of trials required to obtain the initial success in a sequence of Bernoulli trials. In this blog, we will explain the geometric distribution, derive its formula, discuss its mean, and provide examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution that narrates the amount of trials needed to achieve the initial success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment that has two possible results, generally referred to as success and failure. Such as tossing a coin is a Bernoulli trial since it can likewise come up heads (success) or tails (failure).
The geometric distribution is applied when the trials are independent, which means that the outcome of one trial doesn’t affect the outcome of the next trial. In addition, 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 specified 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 first success, k is the number of trials needed to obtain 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 explained as the expected value of the amount of test required to obtain 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 experiments required to achieve the initial success. Such as if the probability of success is 0.5, therefore we anticipate to obtain 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 till the first head turn up.
Let’s assume we flip an honest coin until the first head turns up. The probability of success (obtaining 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 count of coin flips needed to get the first 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 initial head on the first flip is:
P(X = 1) = 0.5^(1-1) * 0.5 = 0.5
For k = 2, the probability of getting the initial 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 first head on the third flip is:
P(X = 3) = 0.5^(3-1) * 0.5 = 0.125
And so forth.
Example 2: Rolling a fair die till the first six appears.
Let’s assume we roll a fair die up until the initial 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 that represents the number of die rolls required to obtain 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 first six on the initial roll is:
P(X = 1) = (5/6)^(1-1) * 1/6 = 1/6
For k = 2, the probability of achieving the first 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 initial 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 important concept in probability theory. It is used to model a broad range of real-world phenomena, such as the count of tests required to obtain the initial success in different situations.
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