Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential department of mathematics which deals with the study of random events. One of the crucial theories in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the number of trials required to obtain the initial success in a sequence of Bernoulli trials. In this blog, we will explain the geometric distribution, derive its formula, discuss its mean, and provide examples.
Explanation of Geometric Distribution
The geometric distribution is a discrete probability distribution which narrates the number of experiments required to achieve the initial success in a succession of Bernoulli trials. A Bernoulli trial is an experiment which has two likely results, generally indicated to as success and failure. Such as tossing a coin is a Bernoulli trial since it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is used when the experiments are independent, which means that the outcome of one experiment does not impact the result of the next test. In addition, the probability of success remains same 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 provided by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which portrays the number of test required to achieve the initial success, k is the count of tests 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 defined as the anticipated value of the number of trials needed to get 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 a single Bernoulli trial.
The mean is the likely count of trials needed to achieve the first success. For instance, if the probability of success is 0.5, therefore we expect to obtain the first success following two trials on average.
Examples of Geometric Distribution
Here are some essential examples of geometric distribution
Example 1: Tossing a fair coin up until the first head turn up.
Suppose we flip a fair coin till the initial head shows up. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is as well as 0.5. Let X be the random variable which represents the count of coin flips required to obtain the initial 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 obtaining 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 obtaining 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 getting 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 first six turns up.
Suppose we roll a fair die till the first 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 irregular variable which depicts the number of die rolls required to achieve 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 achieving the initial six on the first 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 first 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 an essential concept in probability theory. It is used to model a broad range of practical phenomena, such as the count of experiments required to obtain the first success in various scenarios.
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