Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial branch of mathematics which takes up the study of random events. One of the essential ideas in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the amount of trials required to obtain the first success in a sequence of Bernoulli trials. In this article, we will talk about the geometric distribution, derive its formula, discuss its mean, and give examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution that narrates the number of experiments needed to reach the first success in a succession of Bernoulli trials. A Bernoulli trial is an experiment that has two possible results, generally indicated to as success and failure. For instance, tossing a coin is a Bernoulli trial because it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is applied when the experiments are independent, which means that the consequence of one trial does not impact the outcome of the next trial. Additionally, the chances of success remains same throughout all the trials. We can 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 provided by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which represents the number of trials needed to get the first success, k is the number of tests required to obtain the first 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 described as the likely value of the amount of experiments required 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 a single Bernoulli trial.
The mean is the likely number of trials required to get the first success. Such as if the probability of success is 0.5, then we expect to attain the initial success after two trials on average.
Examples of Geometric Distribution
Here are few essential examples of geometric distribution
Example 1: Tossing a fair coin till the first head turn up.
Imagine 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 as well as 0.5. Let X be the random variable that represents the count of coin flips needed to achieve the initial 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 initial head on the first flip is:
P(X = 1) = 0.5^(1-1) * 0.5 = 0.5
For k = 2, the probability of achieving 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 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 until the first six shows up.
Let’s assume we roll a fair die up until the initial six appears. The probability of success (achieving a six) is 1/6, and the probability of failure (obtaining all other number) is 5/6. Let X be the irregular variable that depicts the count of die rolls needed to obtain the initial six. The PMF of X is stated as:
P(X = k) = (1 - 1/6)^(k-1) * 1/6 = (5/6)^(k-1) * 1/6
For k = 1, the probability of getting 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 achieving 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 a crucial theory in probability theory. It is utilized to model a broad range of real-life scenario, for instance the count of experiments required to obtain the first success in various scenarios.
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