Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important department of mathematics which deals with the study of random events. One of the important theories in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the number of experiments required to get the initial success in a sequence of Bernoulli trials. In this blog article, we will define the geometric distribution, derive its formula, discuss its mean, and provide examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution which portrays the number of tests required to achieve the initial success in a succession of Bernoulli trials. A Bernoulli trial is an experiment that has two viable outcomes, typically indicated to as success and failure. Such as tossing a coin is a Bernoulli trial since it can either come up heads (success) or tails (failure).
The geometric distribution is applied when the experiments are independent, meaning that the result of one test doesn’t affect the outcome of the next trial. In addition, the chances of success remains same throughout all the trials. 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 which represents the amount of trials needed to get the initial success, k is the count of trials needed to achieve 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 defined as the anticipated value of the amount of trials required to achieve 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 a single Bernoulli trial.
The mean is the likely count of trials required to get the initial success. For instance, if the probability of success is 0.5, then we expect to get the first 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 till the first 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 which represents the count 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 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 initial 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 forth.
Example 2: Rolling a fair die up until the first six turns up.
Let’s assume we roll a fair die till the initial six shows up. The probability of success (getting a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the irregular variable which depicts the count of die rolls required to achieve the first 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 obtaining 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 forth.
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The geometric distribution is an essential concept in probability theory. It is applied to model a wide range of real-life scenario, for instance the number of tests needed to get the first success in various situations.
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