Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential department of mathematics that deals with the study of random events. One of the essential concepts in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the amount of experiments needed to get the first success in a sequence of Bernoulli trials. In this blog, we will talk about the geometric distribution, derive its formula, discuss its mean, and provide examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the amount of trials required to achieve the first success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment that has two likely outcomes, usually referred to as success and failure. Such as flipping a coin is a Bernoulli trial since it can likewise come up heads (success) or tails (failure).
The geometric distribution is applied when the experiments are independent, which means that the result of one experiment doesn’t affect the result of the next trial. In addition, the chances of success remains same across all the tests. 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 which depicts the amount of trials needed to attain the first success, k is the count of tests needed to achieve 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 anticipated value of the number of experiments required 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 an individual Bernoulli trial.
The mean is the expected number of trials required to obtain the initial success. For instance, if the probability of success is 0.5, therefore we expect to obtain the first success after two trials on average.
Examples of Geometric Distribution
Here are few basic examples of geometric distribution
Example 1: Flipping a fair coin till the first head shows up.
Imagine we toss an honest coin till the first head shows up. The probability of success (obtaining a head) is 0.5, and the probability of failure (obtaining a tail) is also 0.5. Let X be the random variable that depicts the count of coin flips required to get 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 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 first head on the second flip is:
P(X = 2) = 0.5^(2-1) * 0.5 = 0.25
For k = 3, the probability of obtaining 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 initial six appears.
Let’s assume we roll a fair die till the first six appears. The probability of success (getting a six) is 1/6, and the probability of failure (achieving all other number) is 5/6. Let X be the random variable that depicts the number of die rolls needed 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 getting 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 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 getting 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 crucial concept in probability theory. It is utilized to model a wide range of real-world scenario, for example the count of trials required to obtain the first success in various situations.
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