Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important branch of mathematics which handles the study of random occurrence. One of the important concepts in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the number of tests required to get the initial success in a series of Bernoulli trials. In this article, we will explain the geometric distribution, derive its formula, discuss its mean, and offer examples.
Explanation of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the number of trials required to achieve the initial success in a sequence of Bernoulli trials. A Bernoulli trial is a trial that has two likely results, generally indicated to as success and failure. For instance, tossing a coin is a Bernoulli trial because it can either come up heads (success) or tails (failure).
The geometric distribution is utilized when the tests are independent, meaning that the outcome of one trial doesn’t affect the outcome of the upcoming trial. Additionally, the probability of success remains same across all the trials. We could signify 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 amount of trials required to achieve the first success, k is the number of trials required to obtain 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 defined as the likely value of the amount of test needed to achieve the initial 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 likely number of experiments required to get the initial success. For instance, if the probability of success is 0.5, therefore we anticipate to get the first success after two trials on average.
Examples of Geometric Distribution
Here are some essential examples of geometric distribution
Example 1: Tossing a fair coin until the first head shows up.
Let’s assume we flip a fair coin until the initial head turns 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 which represents the number of coin flips needed to achieve the first head. The PMF of X is given by:
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 obtaining the first head on the third flip is:
P(X = 3) = 0.5^(3-1) * 0.5 = 0.125
And so on.
Example 2: Rolling an honest die till the first six turns up.
Let’s assume we roll a fair die up until the first six turns up. The probability of success (getting a six) is 1/6, and the probability of failure (obtaining all other number) is 5/6. Let X be the irregular variable that represents the count of die rolls required to obtain 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 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 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 achieving 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 important theory in probability theory. It is utilized to model a wide range of practical phenomena, such as the number of tests needed to get the first success in different situations.
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