Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important division of mathematics that takes up the study of random occurrence. One of the important concepts in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the number of trials needed to get the initial success in a secession of Bernoulli trials. In this blog article, we will talk about 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 narrates the number of tests needed to reach the initial success in a succession of Bernoulli trials. A Bernoulli trial is an experiment that has two likely 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 applied when the tests are independent, meaning that the consequence of one experiment does not affect the result of the upcoming trial. Additionally, the probability of success remains unchanged throughout all the trials. We could 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 specified by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that represents the number of test needed to achieve the initial success, k is the count of tests needed to obtain 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 defined as the anticipated value of the amount of test needed to get 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 expected number of trials required to get the initial success. For example, if the probability of success is 0.5, then we anticipate to obtain the first success following two trials on average.
Examples of Geometric Distribution
Here are handful of basic examples of geometric distribution
Example 1: Flipping a fair coin up until the first head turn up.
Imagine we toss an honest coin till the first head turns 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 which portrays 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 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 getting 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 first 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 up until the initial six shows up.
Let’s assume we roll a fair die up until 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 represents the number of die rolls required to achieve the initial 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 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 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 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 an essential concept in probability theory. It is utilized to model a wide array of practical scenario, for example the number of trials needed to achieve the first success in several situations.
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