Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important department of math which takes up the study of random occurrence. One of the important concepts in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the number of trials needed to obtain the initial success in a sequence of Bernoulli trials. In this blog, we will talk about the geometric distribution, derive its formula, discuss its mean, and give examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution that narrates the number of tests required to reach the initial success in a sequence of Bernoulli trials. A Bernoulli trial is a trial that has two likely results, usually referred to as success and failure. For example, tossing a coin is a Bernoulli trial because it can likewise come up heads (success) or tails (failure).
The geometric distribution is used when the tests are independent, meaning that the outcome of one test does not affect the outcome of the upcoming trial. In addition, the probability of success remains same throughout 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 portrays the amount of trials needed to get the first success, k is the number of experiments required to attain 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 described as the likely value of the number of trials needed to obtain 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 expected number of tests needed to get the first success. For instance, if the probability of success is 0.5, therefore we expect to get the initial success following two trials on average.
Examples of Geometric Distribution
Here are handful of basic examples of geometric distribution
Example 1: Tossing a fair coin till the first head turn up.
Suppose we flip a fair coin until the first head appears. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is also 0.5. Let X be the random variable which depicts the number of coin flips required to achieve the initial head. The PMF of X is provided as:
P(X = k) = (1 - 0.5)^(k-1) * 0.5 = 0.5^(k-1) * 0.5
For k = 1, the probability of getting 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 initial 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 initial 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 first six appears.
Let’s assume we roll a fair die up until the initial six appears. The probability of success (achieving a six) is 1/6, and the probability of failure (achieving all other number) is 5/6. Let X be the irregular variable that represents the count of die rolls required to obtain 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 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 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 forth.
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The geometric distribution is a crucial theory in probability theory. It is applied to model a wide array of real-world scenario, such as the number of trials needed to obtain the first success in several scenarios.
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