Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is a important branch of mathematics which deals with the study of random occurrence. One of the important theories in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the amount of trials needed to obtain the first success in a sequence of Bernoulli trials. In this blog, we will talk about the geometric distribution, derive its formula, discuss its mean, and give examples.
Explanation of Geometric Distribution
The geometric distribution is a discrete probability distribution which portrays the amount of experiments needed to achieve the first success in a succession of Bernoulli trials. A Bernoulli trial is a trial that has two possible outcomes, generally referred to as success and failure. For instance, tossing a coin is a Bernoulli trial because it can either come up heads (success) or tails (failure).
The geometric distribution is utilized when the tests are independent, meaning that the result of one trial doesn’t impact the result of the upcoming test. Furthermore, the chances of success remains unchanged throughout all the tests. We could 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 specified 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 achieve the initial success, k is the count of tests needed to obtain 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 likely value of the amount of experiments needed 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 expected count of trials required to get the initial success. For instance, if the probability of success is 0.5, then we expect to obtain the first success after two trials on average.
Examples of Geometric Distribution
Here are few primary examples of geometric distribution
Example 1: Tossing a fair coin up until the first head shows up.
Suppose we flip a fair coin till the initial head appears. The probability of success (obtaining a head) is 0.5, and the probability of failure (getting a tail) is also 0.5. Let X be the random variable which represents the count of coin flips required to get 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 achieving 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 getting 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 first six shows up.
Suppose 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 irregular variable that 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 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 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 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 Want from Grade Potential
The geometric distribution is an essential concept in probability theory. It is used to model a wide array of practical scenario, such as the count of experiments needed to obtain the first success in several situations.
If you are having difficulty with probability concepts or any other arithmetic-related subject, Grade Potential Tutoring can guide you. Our adept instructors are available remotely or face-to-face to give personalized and effective tutoring services to support you be successful. Contact us right now to plan a tutoring session and take your math skills to the next level.
