Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial division of mathematics which deals with the study of random occurrence. One of the important theories in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the amount of experiments required to obtain the first success in a series of Bernoulli trials. In this blog, we will explain the geometric distribution, extract its formula, discuss its mean, and provide examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution which narrates the amount of trials required to accomplish the initial success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment that has two viable outcomes, typically indicated to as success and failure. For example, tossing a coin is a Bernoulli trial since it can either come up heads (success) or tails (failure).
The geometric distribution is used when the tests are independent, meaning that the result of one experiment doesn’t affect the outcome of the next test. Additionally, the probability of success remains unchanged across all the trials. We could 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 depicts the number of test required to attain the first success, k is the count of trials needed to attain 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 likely value of the amount of test needed to achieve 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 a single Bernoulli trial.
The mean is the anticipated count of experiments needed to obtain the first 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 handful of essential examples of geometric distribution
Example 1: Tossing a fair coin until the first head turn up.
Let’s assume we flip an honest coin until the first head appears. 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 obtain the first 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 first 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 first head on the second flip is:
P(X = 2) = 0.5^(2-1) * 0.5 = 0.25
For k = 3, the probability of obtaining 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 an honest die until the initial six shows up.
Let’s assume we roll a fair die up until the first six appears. The probability of success (obtaining a six) is 1/6, and the probability of failure (obtaining any other number) is 5/6. Let X be the irregular variable which depicts the count of die rolls required to achieve the first six. The PMF of X is stated 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 getting 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 getting 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 important theory in probability theory. It is used to model a wide array of real-world scenario, for example the number of experiments needed to achieve the first success in various situations.
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