Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial division of math which 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 which models the amount of tests required to get the initial success in a secession of Bernoulli trials. In this article, we will define the geometric distribution, extract its formula, discuss its mean, and give examples.
Meaning of Geometric Distribution
The geometric distribution is a discrete probability distribution that describes the number of tests required to accomplish the first success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment which has two viable outcomes, usually 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 result of one test does not impact the outcome of the upcoming test. Furthermore, the chances of success remains same 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 given by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that depicts the number of trials required to get the initial success, k is the number of tests required to attain the initial 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 likely value of the number of experiments needed to obtain 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 anticipated number of experiments required to get the initial success. For instance, 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 few essential examples of geometric distribution
Example 1: Flipping a fair coin up until the first head shows up.
Suppose we flip a fair coin till the initial head turns up. The probability of success (obtaining 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 which depicts the number of coin flips required to get 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 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 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 a fair die till the first six shows up.
Suppose we roll an honest die up until the first six appears. The probability of success (obtaining a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the random variable that portrays the number of die rolls needed to get 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 getting 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 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 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 an essential concept in probability theory. It is used to model a broad range of real-life scenario, for instance the number of tests needed to achieve the initial success in various scenarios.
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