Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial division of math that takes up the study of random occurrence. One of the crucial ideas in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the number of trials required to obtain the initial 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.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution that portrays the number of trials needed to accomplish the first success in a sequence of Bernoulli trials. A Bernoulli trial is an experiment that has two possible results, usually indicated to as success and failure. For example, tossing a coin is a Bernoulli trial since it can likewise come up heads (success) or tails (failure).
The geometric distribution is utilized when the trials are independent, which means that the consequence of one test doesn’t affect the result of the next trial. Furthermore, the chances of success remains constant across all the trials. We could signify 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 represents the number of test required to attain the initial success, k is the number of experiments required to obtain the first 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 described as the likely value of the amount of trials required to obtain 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 a single Bernoulli trial.
The mean is the likely number of tests needed to obtain the first success. For example, if the probability of success is 0.5, therefore we expect to get the first success after two trials on average.
Examples of Geometric Distribution
Here are handful of basic examples of geometric distribution
Example 1: Flipping a fair coin up until the first head shows up.
Suppose we flip an honest coin till the first head shows 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 that depicts the count of coin flips needed 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 first head on the first flip is:
P(X = 1) = 0.5^(1-1) * 0.5 = 0.5
For k = 2, the probability of achieving 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 forth.
Example 2: Rolling a fair die up until the initial six shows up.
Suppose we roll an honest die until the first six turns up. The probability of success (achieving a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the random variable which portrays the number of die rolls required to achieve the initial 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 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 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 getting 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.
Get the Tutoring You Need from Grade Potential
The geometric distribution is a important concept in probability theory. It is utilized to model a wide array of real-life scenario, such as the number of trials required to get the initial success in several scenarios.
If you are feeling challenged with probability concepts or any other arithmetic-related topic, Grade Potential Tutoring can support you. Our experienced teachers are accessible remotely or face-to-face to provide customized and productive tutoring services to guide you succeed. Connect with us today to schedule a tutoring session and take your math skills to the next level.
