Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial division of mathematics that handles the study of random occurrence. One of the essential theories in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the amount of trials required to obtain the first success in a series of Bernoulli trials. In this blog, we will define the geometric distribution, extract its formula, discuss its mean, and offer examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution which portrays the number of trials needed to achieve the initial success in a series of Bernoulli trials. A Bernoulli trial is a trial that has two likely results, typically indicated to as success and failure. For example, flipping a coin is a Bernoulli trial because it can either come up heads (success) or tails (failure).
The geometric distribution is used when the trials are independent, meaning that the consequence of one experiment doesn’t affect the result of the next trial. Furthermore, 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 provided by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that portrays the number of trials needed to attain the initial success, k is the number of experiments required to achieve the initial success, p is the probability of success in a single 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 experiments needed to get 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 a single Bernoulli trial.
The mean is the likely count of experiments required to get the first success. For instance, if the probability of success is 0.5, then we expect to get the first success following two trials on average.
Examples of Geometric Distribution
Here are some basic examples of geometric distribution
Example 1: Tossing a fair coin up until the first head shows up.
Let’s assume we flip an honest coin until the initial head shows up. The probability of success (obtaining a head) is 0.5, and the probability of failure (getting a tail) is also 0.5. Let X be the random variable which portrays the number of coin flips required 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 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 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 achieving the initial head on the third flip is:
P(X = 3) = 0.5^(3-1) * 0.5 = 0.125
And so on.
Example 2: Rolling a fair die till the first six shows up.
Suppose we roll a fair die until the initial six appears. 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 that represents the number of die rolls required to get the first 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 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 getting the first 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 forth.
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The geometric distribution is an essential theory in probability theory. It is utilized to model a wide range of real-world phenomena, for example the count of tests needed to achieve the first success in different situations.
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