Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important department of mathematics which handles the study of random occurrence. One of the crucial ideas in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the amount of tests required to obtain the first success in a series of Bernoulli trials. In this article, we will talk about the geometric distribution, derive its formula, discuss its mean, and offer examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the number of experiments required to achieve the first success in a sequence of Bernoulli trials. A Bernoulli trial is a test which has two viable results, generally indicated to as success and failure. For instance, flipping a coin is a Bernoulli trial since it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is applied when the tests are independent, which means that the outcome of one test does not impact the outcome of the upcoming test. Additionally, the chances of success remains same across all the tests. We could 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 given by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that portrays the amount of trials needed to get the initial success, k is the count of trials needed to attain 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 likely value of the amount of experiments needed to get the first success. The mean is given by 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 get the initial success. Such as if the probability of success is 0.5, therefore we expect to get the initial success following two trials on average.
Examples of Geometric Distribution
Here are handful of basic examples of geometric distribution
Example 1: Flipping a fair coin up until the first head appears.
Suppose we toss a fair coin until the initial head turns up. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is also 0.5. Let X be the random variable which represents the number of coin flips needed to achieve the initial 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 achieving 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 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 initial 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 until the first six appears.
Let’s assume we roll an honest die up until the initial six appears. The probability of success (achieving a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the random variable that depicts the count of die rolls required to obtain 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 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 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 on.
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The geometric distribution is a crucial theory in probability theory. It is used to model a wide range of practical scenario, for instance the number of tests required to obtain the first success in several situations.
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