Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important division of mathematics 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 distinct probability distribution which models the number of tests required to get the first success in a sequence of Bernoulli trials. In this article, we will talk about the geometric distribution, extract its formula, discuss its mean, and provide examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the amount of experiments needed to accomplish the initial success in a series of Bernoulli trials. A Bernoulli trial is a trial that has two likely outcomes, typically referred to as success and failure. For instance, tossing a coin is a Bernoulli trial since it can either turn out to be heads (success) or tails (failure).
The geometric distribution is used when the trials are independent, meaning that the consequence of one test does not affect the outcome of the next trial. In addition, the chances of success remains constant throughout all the tests. We could 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 represents the number of test required to attain the first success, k is the number of trials needed to achieve 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 described as the likely value of the number of test required to obtain 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 expected count of experiments needed to get the first success. For example, if the probability of success is 0.5, therefore we anticipate to get the first success after two trials on average.
Examples of Geometric Distribution
Here are few essential examples of geometric distribution
Example 1: Flipping a fair coin until the first head turn up.
Let’s assume we flip a fair coin till the initial head shows up. The probability of success (obtaining 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 portrays the count of coin flips needed to get the first 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 achieving 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 achieving the initial head on the third flip is:
P(X = 3) = 0.5^(3-1) * 0.5 = 0.125
And so on.
Example 2: Rolling a fair die up until the initial six shows up.
Suppose we roll an honest die till the initial six shows up. 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 that depicts the count of die rolls needed to get the initial six. The PMF of X is given by:
P(X = k) = (1 - 1/6)^(k-1) * 1/6 = (5/6)^(k-1) * 1/6
For k = 1, the probability of achieving 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 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 important theory in probability theory. It is applied to model a wide range of practical phenomena, for example the count of trials needed to achieve the initial success in different situations.
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