Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial department of mathematics that 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 number of experiments needed to obtain the first success in a series of Bernoulli trials. In this article, we will define the geometric distribution, extract its formula, discuss its mean, and give examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the amount of tests needed to accomplish the initial success in a succession of Bernoulli trials. A Bernoulli trial is a trial which has two possible results, usually indicated to as success and failure. Such as tossing a coin is a Bernoulli trial because it can either come up heads (success) or tails (failure).
The geometric distribution is used when the trials are independent, which means that the consequence of one test does not affect the outcome of the upcoming trial. Furthermore, the chances of success remains constant across all the tests. 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 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 attain the initial success, k is the number of experiments required 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 likely value of the number of test 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 expected count of tests needed to achieve the first success. Such as if the probability of success is 0.5, then we anticipate to attain the initial success following two trials on average.
Examples of Geometric Distribution
Here are some basic examples of geometric distribution
Example 1: Tossing a fair coin until the first head turn up.
Let’s assume we flip a fair coin till the initial head turns up. The probability of success (obtaining 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 needed to achieve 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 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 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 forth.
Example 2: Rolling an honest die until the first six appears.
Suppose we roll an honest die until the initial six turns up. The probability of success (obtaining 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 count of die rolls required to get 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 getting the first six on the first 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 achieving 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 applied to model a wide range of real-life phenomena, such as the number of tests required to obtain the first success in various situations.
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