Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important branch of mathematics which takes up the study of random occurrence. One of the essential concepts in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the amount of tests needed to obtain the initial success in a secession of Bernoulli trials. In this blog article, we will talk about the geometric distribution, extract its formula, discuss its mean, and offer examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution that narrates the amount of tests required to reach the first success in a succession of Bernoulli trials. A Bernoulli trial is a trial which has two possible results, typically indicated to as success and failure. Such as flipping a coin is a Bernoulli trial because it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is utilized when the experiments are independent, which means that the outcome of one experiment does not affect the result of the upcoming trial. Additionally, the chances of success remains constant across all the tests. 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 provided by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which portrays the number of trials needed to achieve the first success, k is the number of trials 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 explained as the anticipated value of the amount of test needed to obtain the initial 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 likely count of trials needed to get the initial success. For example, if the probability of success is 0.5, therefore we expect to get the first success following two trials on average.
Examples of Geometric Distribution
Here are few basic examples of geometric distribution
Example 1: Flipping a fair coin up until the first head appears.
Suppose we toss an honest coin until the first head turns up. The probability of success (getting a head) is 0.5, and the probability of failure (getting a tail) is as well as 0.5. Let X be the random variable which depicts the number of coin flips required to achieve the first 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 obtaining 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 obtaining 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 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 till the initial six appears.
Suppose we roll an honest die up until the first six turns up. The probability of success (getting a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the random variable that depicts the number of die rolls required to obtain 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 first roll is:
P(X = 1) = (5/6)^(1-1) * 1/6 = 1/6
For k = 2, the probability of achieving 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 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 on.
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The geometric distribution is a important theory in probability theory. It is utilized to model a wide array of real-world phenomena, for example the count of trials needed to get the initial success in several situations.
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