Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential department of mathematics which deals with the study of random events. One of the crucial ideas in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the number of tests needed to get the initial success in a series of Bernoulli trials. In this blog article, we will define the geometric distribution, derive its formula, discuss its mean, and give examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution that narrates the number of trials needed to reach the initial success in a series of Bernoulli trials. A Bernoulli trial is an experiment that has two viable outcomes, typically referred to as success and failure. For instance, tossing a coin is a Bernoulli trial since it can likewise come up heads (success) or tails (failure).
The geometric distribution is utilized when the experiments are independent, which means that the result of one experiment does not impact the outcome of the next test. Furthermore, the probability of success remains same across all the tests. 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 that depicts the amount of trials needed to achieve the first success, k is the number of trials needed to attain the first 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 expected value of the number of test required to get the first 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 required to get the first success. Such as if the probability of success is 0.5, then we anticipate to attain the initial success after two trials on average.
Examples of Geometric Distribution
Here are handful of essential examples of geometric distribution
Example 1: Tossing a fair coin till the first head appears.
Let’s assume we flip an honest coin till the initial head shows up. The probability of success (getting a head) is 0.5, and the probability of failure (getting a tail) is also 0.5. Let X be the random variable that represents the number of coin flips needed to achieve the first head. The PMF of X is given by:
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 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 getting 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 an honest die until the initial six shows up. The probability of success (getting a six) is 1/6, and the probability of failure (achieving all other number) is 5/6. Let X be the random variable which portrays the number 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 achieving 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 an essential theory in probability theory. It is used to model a wide range of real-world phenomena, for example the number of tests needed to obtain the first success in various scenarios.
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