Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important division of math that deals with the study of random events. One of the important theories 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 secession of Bernoulli trials. In this article, we will talk about the geometric distribution, extract its formula, discuss its mean, and offer examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution that portrays the number of experiments needed to accomplish the first success in a succession of Bernoulli trials. A Bernoulli trial is an experiment that has two viable results, typically indicated to as success and failure. Such as 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 trials are independent, which means that the consequence of one test doesn’t impact the result of the upcoming trial. Additionally, the probability of success remains same across all the tests. 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 given by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which portrays the amount of trials needed to achieve the first success, k is the count of experiments required to obtain the initial 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 defined as the anticipated value of the number of trials required to get the first 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 likely number of experiments needed to get the initial success. For example, if the probability of success is 0.5, therefore we expect to attain the first success following two trials on average.
Examples of Geometric Distribution
Here are few basic examples of geometric distribution
Example 1: Flipping a fair coin till the first head appears.
Let’s assume we flip a fair coin until the initial head shows up. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is as well as 0.5. Let X be the random variable that depicts the number of coin flips needed to achieve the initial head. The PMF of X is stated as:
P(X = k) = (1 - 0.5)^(k-1) * 0.5 = 0.5^(k-1) * 0.5
For k = 1, the probability of getting 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 achieving 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 getting 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 up until the first six appears.
Let’s assume we roll an honest die until the first six appears. The probability of success (achieving a six) is 1/6, and the probability of failure (getting any other number) is 5/6. Let X be the irregular variable which portrays the count of die rolls needed 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 obtaining 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 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 achieving 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 concept in probability theory. It is applied to model a broad range of practical phenomena, such as the number of tests required to obtain the initial success in various situations.
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