Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential department of math which takes up the study of random occurrence. One of the essential ideas in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the amount of experiments needed to obtain the initial success in a secession of Bernoulli trials. In this article, we will explain the geometric distribution, extract its formula, discuss its mean, and offer examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution which narrates the amount of trials required to achieve the first success in a sequence of Bernoulli trials. A Bernoulli trial is a trial that has two possible results, usually referred 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 utilized when the tests are independent, meaning that the result of one experiment doesn’t impact the outcome of the next trial. Additionally, the probability of success remains constant throughout 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 attain 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 expected value of the number of test needed to obtain the first success. The mean is stated in 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 number of experiments needed to achieve the initial success. For example, if the probability of success is 0.5, therefore we anticipate to obtain the initial success after two trials on average.
Examples of Geometric Distribution
Here are some basic examples of geometric distribution
Example 1: Tossing a fair coin up until the first head turn up.
Let’s assume we flip a fair coin till the initial head turns up. The probability of success (getting a head) is 0.5, and the probability of failure (getting a tail) is also 0.5. Let X be the random variable that depicts the count of coin flips required to get the initial 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 getting 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 initial 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 till the initial six turns up.
Suppose we roll a fair die until the initial six appears. The probability of success (getting a six) is 1/6, and the probability of failure (obtaining all other number) is 5/6. Let X be the irregular variable which represents the number of die rolls required to get the first 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 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 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 initial 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 an essential theory in probability theory. It is applied to model a broad range of real-life phenomena, for example the count of tests required to achieve the initial success in several scenarios.
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