Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial branch of mathematics that deals with the study of random events. One of the essential ideas in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the amount of trials required to obtain the first success in a series of Bernoulli trials. In this blog article, we will talk about the geometric distribution, derive its formula, discuss its mean, and offer examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution that portrays the number of trials required to reach the first success in a sequence of Bernoulli trials. A Bernoulli trial is a trial that has two viable outcomes, generally referred 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 used when the experiments are independent, which means that the outcome of one trial does not affect the outcome of the next test. In addition, the chances 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 specified by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which portrays the amount of test needed to get the first success, k is the count of experiments 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 explained as the anticipated value of the number of experiments required 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 an individual Bernoulli trial.
The mean is the anticipated number of experiments needed to achieve the initial success. For instance, if the probability of success is 0.5, then we anticipate to obtain the initial success following two trials on average.
Examples of Geometric Distribution
Here are handful of primary examples of geometric distribution
Example 1: Flipping a fair coin until the first head appears.
Imagine we flip a fair coin till the initial head turns up. The probability of success (getting a head) is 0.5, and the probability of failure (getting a tail) is as well as 0.5. Let X be the random variable that depicts the count of coin flips required to obtain 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 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 appears.
Suppose we roll an honest die until the first six appears. The probability of success (obtaining a six) is 1/6, and the probability of failure (obtaining all other number) is 5/6. Let X be the random variable that portrays the count of die rolls needed to get the first 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 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 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 on.
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The geometric distribution is a important concept in probability theory. It is utilized to model a wide array of real-life phenomena, for example the count of tests required to obtain the first success in several situations.
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