Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential department of math which takes up the study of random occurrence. One of the essential 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 initial success in a secession of Bernoulli trials. In this article, we will define the geometric distribution, derive its formula, discuss its mean, and offer examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution which describes the amount of trials needed to reach the first success in a succession of Bernoulli trials. A Bernoulli trial is an experiment that has two viable outcomes, usually referred to as success and failure. Such as tossing a coin is a Bernoulli trial because it can either turn out to be heads (success) or tails (failure).
The geometric distribution is used when the tests are independent, which means that the result of one trial doesn’t affect the outcome of the upcoming trial. Furthermore, the probability of success remains unchanged across all the tests. 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 required to get the initial 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 expected value of the number of experiments required 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 count of experiments needed to achieve the first success. For example, if the probability of success is 0.5, then we anticipate to attain the first 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 till the first head appears.
Suppose we flip an honest coin till the first head appears. 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 which depicts the number of coin flips needed to obtain 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 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 obtaining the initial 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 first 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 till the first six turns up.
Suppose we roll an honest die up until the initial six turns up. The probability of success (getting a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the irregular variable that 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 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 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 a crucial theory in probability theory. It is used to model a wide range of real-life phenomena, for example the count of experiments required to achieve the initial success in various situations.
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