Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential department of mathematics that handles the study of random occurrence. One of the important theories 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 article, we will define the geometric distribution, derive its formula, discuss its mean, and provide examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the number of experiments required to accomplish the initial success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment which has two likely outcomes, typically indicated to as success and failure. For example, 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, meaning that the result of one trial doesn’t impact the outcome of the next test. Additionally, the chances of success remains constant 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 given by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that depicts the number of test needed to get the first success, k is the number of trials required 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 expected value of the amount of trials needed to obtain the first 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 obtain the initial success. For instance, if the probability of success is 0.5, therefore we expect to attain the initial success after two trials on average.
Examples of Geometric Distribution
Here are some basic examples of geometric distribution
Example 1: Flipping a fair coin until the first head turn up.
Suppose we flip an honest coin till the first head shows up. 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 that depicts the count of coin flips needed to obtain 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 first head on the first flip is:
P(X = 1) = 0.5^(1-1) * 0.5 = 0.5
For k = 2, the probability of getting 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 an honest die up until the initial six appears.
Suppose we roll an honest die up until the initial six appears. The probability of success (achieving a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the random variable which depicts the number of die rolls required to get 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 achieving 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 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 obtaining 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 an essential concept in probability theory. It is applied to model a broad array of real-life phenomena, for example the number of trials required to achieve the initial success in several scenarios.
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