Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important division of mathematics that takes up 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 get the first success in a sequence of Bernoulli trials. In this blog article, we will talk about the geometric distribution, derive its formula, discuss its mean, and provide examples.
Explanation of Geometric Distribution
The geometric distribution is a discrete probability distribution that narrates the amount of experiments required to achieve the initial success in a series of Bernoulli trials. A Bernoulli trial is a trial that has two likely outcomes, typically indicated to as success and failure. Such as flipping a coin is a Bernoulli trial because it can likewise come up heads (success) or tails (failure).
The geometric distribution is applied when the trials are independent, meaning that the consequence of one trial does not impact the outcome of the upcoming trial. Additionally, 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 amount of test required to get the initial success, k is the count of experiments required to achieve the first 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 an individual Bernoulli trial.
The mean is the anticipated count of experiments needed to obtain the initial 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 some primary examples of geometric distribution
Example 1: Tossing a fair coin until the first head appears.
Suppose we flip an honest coin till the first head appears. 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 number of coin flips required to obtain the initial head. The PMF of X is provided 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 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 till the initial six shows up.
Let’s assume we roll a fair die till the first six turns 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 irregular variable which represents the number of die rolls required to obtain the first six. The PMF of X is provided as:
P(X = k) = (1 - 1/6)^(k-1) * 1/6 = (5/6)^(k-1) * 1/6
For k = 1, the probability of achieving 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 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 obtaining 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 important theory in probability theory. It is utilized to model a wide range of practical phenomena, for example the number of trials required to obtain the first success in various scenarios.
If you are feeling challenged with probability theory or any other mathematics-related topic, Grade Potential Tutoring can guide you. Our expert tutors are accessible online or face-to-face to give personalized and productive tutoring services to help you succeed. Contact us right now to schedule a tutoring session and take your math abilities to the next level.
