Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial branch of mathematics which takes up the study of random events. One of the essential ideas 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 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 that portrays the number of experiments needed to accomplish the initial success in a succession of Bernoulli trials. A Bernoulli trial is an experiment that has two likely results, generally referred to as success and failure. For instance, flipping a coin is a Bernoulli trial because it can likewise come up heads (success) or tails (failure).
The geometric distribution is utilized when the trials are independent, which means that the result of one experiment doesn’t impact the outcome of the upcoming test. Furthermore, the chances 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 specified by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that represents the number of trials required to attain the initial success, k is the number of tests needed to achieve 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 described as the likely value of the number of experiments required 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 an individual Bernoulli trial.
The mean is the anticipated number of trials required to obtain the first success. For example, if the probability of success is 0.5, then we anticipate to obtain the initial success following two trials on average.
Examples of Geometric Distribution
Here are some essential examples of geometric distribution
Example 1: Flipping a fair coin until the first head appears.
Let’s assume we flip a fair coin till the first head shows 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 that 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 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 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 achieving 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 a fair die until the initial six appears.
Let’s assume we roll an honest die till the initial 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 that portrays the count of die rolls required to achieve the first 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 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 achieving 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 first 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 theory in probability theory. It is applied to model a wide range of real-life phenomena, for example the number of trials needed to obtain the initial success in several situations.
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