Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important division of math which deals with the study of random occurrence. One of the essential theories in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the number of trials required to obtain the initial success in a secession of Bernoulli trials. In this blog, we will explain 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 experiments required to accomplish the initial success in a succession of Bernoulli trials. A Bernoulli trial is a trial which has two likely results, typically indicated to as success and failure. Such as tossing a coin is a Bernoulli trial since it can either turn out to be heads (success) or tails (failure).
The geometric distribution is applied when the experiments are independent, which means that the result of one test doesn’t impact the outcome of the next test. Additionally, the chances of success remains same across 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 provided by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable which portrays the number of trials required to attain the initial success, k is the count of trials required to obtain the initial 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 defined as the expected 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 likely count of experiments required to achieve the first success. For example, if the probability of success is 0.5, then we anticipate 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 until the first head appears.
Imagine we flip an honest coin until the first head shows up. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is also 0.5. Let X be the random variable which represents the count of coin flips needed to get the initial 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 obtaining 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 first 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 up until the first six shows up.
Let’s assume we roll a fair die up until the initial six appears. The probability of success (getting a six) is 1/6, and the probability of failure (getting any other number) is 5/6. Let X be the random variable which portrays the number of die rolls required 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 first 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 achieving the first 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 utilized to model a wide array of real-life scenario, such as the number of experiments required to obtain the initial success in several situations.
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