Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential branch 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 distinct probability distribution which models the number of trials needed to obtain the initial success in a sequence of Bernoulli trials. In this blog article, we will talk about the geometric distribution, derive its formula, discuss its mean, and provide examples.
Explanation of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the amount of tests required to achieve the first success in a succession of Bernoulli trials. A Bernoulli trial is an experiment which has two likely results, typically referred to as success and failure. For example, 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 result of one experiment doesn’t affect the outcome of the upcoming test. Additionally, the probability of success remains unchanged throughout all the trials. We could denote 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 that portrays the number of test required to get the first success, k is the count of tests needed to obtain the initial 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 achieve 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 an individual Bernoulli trial.
The mean is the likely number of tests required to achieve the initial success. For instance, if the probability of success is 0.5, then we expect to attain the first 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 up until the first head appears.
Imagine we flip a fair coin till the initial 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 that represents 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 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 up until the initial six turns up.
Suppose we roll an honest die until the first six shows up. The probability of success (getting a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the random variable which represents 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 obtaining 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 getting 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 getting the first 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 crucial concept in probability theory. It is utilized to model a broad array of real-life scenario, such as the count of tests required to achieve the first success in different situations.
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