Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential division of mathematics that takes up the study of random events. One of the important ideas in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the number of tests required to get the initial success in a secession of Bernoulli trials. In this 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 narrates the number of trials required to achieve the first success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment which has two possible outcomes, usually referred to as success and failure. Such as flipping a coin is a Bernoulli trial since it can likewise come up heads (success) or tails (failure).
The geometric distribution is applied when the experiments are independent, which means that the outcome of one experiment does not impact the outcome of the next trial. Furthermore, the chances of success remains same throughout 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 which depicts the amount of test required to attain the first success, k is the number of trials required to obtain 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 described as the anticipated value of the amount of trials needed to get 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 count of trials required to obtain the first success. For instance, if the probability of success is 0.5, then we anticipate to obtain the first 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 until the first head appears.
Suppose we flip a fair coin till the first head appears. 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 portrays the number of coin flips needed to achieve 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 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 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 an honest die up until the initial six shows up.
Suppose we roll a fair die up until the initial six turns up. The probability of success (obtaining a six) is 1/6, and the probability of failure (obtaining any other number) is 5/6. Let X be the random variable which depicts the count of die rolls needed 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 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 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 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 forth.
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The geometric distribution is an essential concept in probability theory. It is applied to model a broad array of real-life phenomena, such as the number of tests required to get the first success in various scenarios.
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