Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important division of mathematics which handles the study of random events. One of the important ideas in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution which models the amount of tests required to obtain the first success in a sequence of Bernoulli trials. In this blog article, we will define the geometric distribution, derive its formula, discuss its mean, and offer examples.
Explanation of Geometric Distribution
The geometric distribution is a discrete probability distribution that portrays the number of trials required to accomplish the initial success in a series of Bernoulli trials. A Bernoulli trial is a trial that has two likely results, generally referred to as success and failure. For instance, 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 experiments are independent, meaning that the outcome of one test doesn’t impact the result of the next test. Furthermore, the chances of success remains same across all the tests. 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 that represents the number of trials required to achieve the initial success, k is the number of tests needed to attain 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 explained as the anticipated value of the number of experiments required 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 anticipated count of experiments required to achieve the initial success. For example, if the probability of success is 0.5, then we anticipate to get the first success after two trials on average.
Examples of Geometric Distribution
Here are some basic examples of geometric distribution
Example 1: Flipping a fair coin till the first head appears.
Imagine we toss an honest coin till 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 that depicts the number of coin flips required to get the first 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 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 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 obtaining 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 until the initial six turns up.
Suppose we roll an honest die till the initial six shows up. The probability of success (obtaining a six) is 1/6, and the probability of failure (obtaining any other number) is 5/6. Let X be the random variable which represents the count of die rolls needed 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 obtaining 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 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 on.
Get the Tutoring You Require from Grade Potential
The geometric distribution is a important concept in probability theory. It is used to model a wide range of practical phenomena, for example the number of tests needed to obtain the initial success in different scenarios.
If you are struggling with probability concepts or any other math-related subject, Grade Potential Tutoring can support you. Our experienced instructors are available remotely or in-person to give customized and effective tutoring services to help you succeed. Call us right now to plan a tutoring session and take your math abilities to the next level.
