Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial department of mathematics that deals with the study of random occurrence. One of the important ideas in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the amount of trials required to get the first success in a series of Bernoulli trials. In this blog, we will explain 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 number of trials required to accomplish the initial success in a sequence of Bernoulli trials. A Bernoulli trial is a test which has two likely results, typically indicated to as success and failure. For example, flipping a coin is a Bernoulli trial because it can either turn out to be heads (success) or tails (failure).
The geometric distribution is utilized when the trials 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 constant across all the trials. 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 provided by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which depicts the number of trials needed to attain the first success, k is the number of trials needed 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 test required to achieve 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 count of tests needed to get the first success. For instance, if the probability of success is 0.5, therefore we expect to attain the first success following two trials on average.
Examples of Geometric Distribution
Here are some basic examples of geometric distribution
Example 1: Tossing a fair coin up until the first head shows up.
Let’s assume we flip an honest coin until the first head turns up. The probability of success (obtaining a head) is 0.5, and the probability of failure (obtaining a tail) is as well as 0.5. Let X be the random variable that portrays 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 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 an honest die up until the first six shows up.
Let’s assume we roll a fair die until the initial six shows up. 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 represents the count 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 forth.
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The geometric distribution is a important theory in probability theory. It is applied to model a broad range of real-world scenario, for example the number of experiments required to obtain the first success in different situations.
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