Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important department of mathematics 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 that models the amount of experiments needed to get the first success in a secession of Bernoulli trials. In this blog article, 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 which portrays the amount of tests required to accomplish the initial success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment that has two likely outcomes, typically indicated to as success and failure. For instance, flipping a coin is a Bernoulli trial because it can either come up heads (success) or tails (failure).
The geometric distribution is applied when the experiments are independent, which means that the outcome of one trial doesn’t affect the outcome of the upcoming test. Furthermore, the chances of success remains unchanged throughout all the trials. We can indicate 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 depicts the number of test needed to attain the first success, k is the count of trials needed to obtain 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 likely value of the amount of test needed to get the initial 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 tests required to obtain the first success. Such as if the probability of success is 0.5, then we anticipate to get the initial success following two trials on average.
Examples of Geometric Distribution
Here are some essential examples of geometric distribution
Example 1: Tossing a fair coin until the first head appears.
Let’s assume we flip a fair coin until the first head turns 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 depicts the count of coin flips required to achieve the initial head. The PMF of X is stated as:
P(X = k) = (1 - 0.5)^(k-1) * 0.5 = 0.5^(k-1) * 0.5
For k = 1, the probability of obtaining 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 first 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 initial 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 till the initial six turns up.
Suppose we roll a fair die until the initial six shows up. The probability of success (achieving a six) is 1/6, and the probability of failure (achieving all other number) is 5/6. Let X be the random variable which represents the number of die rolls required to get the first 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 first roll is:
P(X = 1) = (5/6)^(1-1) * 1/6 = 1/6
For k = 2, the probability of getting 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.
Get the Tutoring You Require from Grade Potential
The geometric distribution is a crucial theory in probability theory. It is used to model a wide range of real-life phenomena, for instance the count of trials required to obtain the initial success in various situations.
If you are feeling challenged with probability concepts or any other arithmetic-related subject, Grade Potential Tutoring can guide you. Our adept instructors are accessible remotely or in-person to offer customized and effective tutoring services to guide you succeed. Contact us right now to plan a tutoring session and take your math abilities to the next stage.
