Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential division 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 distinct probability distribution that models the amount of experiments needed to obtain the initial success in a series of Bernoulli trials. In this blog article, we will define the geometric distribution, derive its formula, discuss its mean, and give examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution which describes the amount of tests needed to accomplish the first success in a series of Bernoulli trials. A Bernoulli trial is a trial that has two viable results, generally indicated to as success and failure. For example, flipping a coin is a Bernoulli trial because 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 experiment does not affect the result of the next trial. Furthermore, the probability of success remains same throughout all the trials. We can signify 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 portrays the amount of test needed to attain the first success, k is the number of trials needed to obtain 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 anticipated value of the number of experiments required to achieve 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 an individual Bernoulli trial.
The mean is the likely number of tests required to get the initial success. For instance, if the probability of success is 0.5, therefore we anticipate to attain the initial success after two trials on average.
Examples of Geometric Distribution
Here are handful of basic examples of geometric distribution
Example 1: Tossing a fair coin until the first head turn up.
Suppose we flip an honest 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 required to obtain 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 achieving 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 initial 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 initial 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 shows up. 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 random variable which portrays the count of die rolls needed to get 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 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 getting 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 getting 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 a important concept in probability theory. It is utilized to model a wide range of real-life phenomena, for instance the count of trials required to get the first success in several situations.
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