Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial division of math that takes up 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 number of tests required to obtain the initial success in a sequence of Bernoulli trials. In this blog article, we will talk about the geometric distribution, extract its formula, discuss its mean, and give examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution which describes the amount of tests needed to reach the first success in a series of Bernoulli trials. A Bernoulli trial is an experiment that has two possible outcomes, typically indicated to as success and failure. For instance, tossing a coin is a Bernoulli trial since it can either turn out to be heads (success) or tails (failure).
The geometric distribution is used when the trials are independent, which means that the result of one experiment does not affect the outcome of the next trial. Furthermore, the probability of success remains unchanged across all the tests. We can 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 required to achieve the initial success, k is the number of trials required 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 explained as the anticipated value of the amount of trials needed 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 a single Bernoulli trial.
The mean is the expected number of tests required to get the first success. For example, if the probability of success is 0.5, then we anticipate to obtain the initial success after two trials on average.
Examples of Geometric Distribution
Here are handful of basic examples of geometric distribution
Example 1: Flipping a fair coin up until the first head appears.
Suppose we flip a fair coin until 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 number of coin flips needed to obtain the first 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 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 getting 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 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 till the first six turns up.
Suppose we roll a fair die until the initial six turns up. The probability of success (obtaining a six) is 1/6, and the probability of failure (getting any other number) is 5/6. Let X be the random variable which portrays the count of die rolls required to obtain the initial six. The PMF of X is provided 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 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 important concept in probability theory. It is applied to model a wide array of practical phenomena, for instance the number of trials required to achieve the initial success in several scenarios.
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