Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential branch of math which takes up the study of random events. One of the crucial concepts in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution which models the amount of experiments needed to obtain the first success in a sequence of Bernoulli trials. In this blog, we will talk about 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 narrates the number of experiments required to accomplish the first success in a succession of Bernoulli trials. A Bernoulli trial is a trial which has two likely outcomes, usually indicated to as success and failure. Such as tossing a coin is a Bernoulli trial since it can either come up heads (success) or tails (failure).
The geometric distribution is used when the experiments are independent, meaning that the consequence of one trial does not impact the outcome of the next trial. Additionally, the chances of success remains constant throughout 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 which represents the number of test needed to get the initial success, k is the number of tests 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 described as the expected value of the amount of experiments required to get 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 likely count of trials required to get the first success. For example, if the probability of success is 0.5, then we anticipate to get the initial success after two trials on average.
Examples of Geometric Distribution
Here are some essential examples of geometric distribution
Example 1: Flipping a fair coin till the first head shows up.
Let’s assume we toss an honest coin until the initial head shows up. The probability of success (obtaining a head) is 0.5, and the probability of failure (getting a tail) is as well as 0.5. Let X be the random variable which portrays the number of coin flips needed 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 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 first 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 up until the initial six turns up.
Suppose we roll a fair die till the initial six appears. The probability of success (obtaining a six) is 1/6, and the probability of failure (achieving any other number) is 5/6. Let X be the irregular variable that represents the number of die rolls needed to obtain 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 getting the initial six on the initial roll is:
P(X = 1) = (5/6)^(1-1) * 1/6 = 1/6
For k = 2, the probability of achieving 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 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 Want from Grade Potential
The geometric distribution is a important theory in probability theory. It is applied to model a broad range of real-life scenario, such as the count of experiments needed to get the initial success in different situations.
If you are feeling challenged with probability concepts or any other math-related topic, Grade Potential Tutoring can support you. Our expert teachers are available online or face-to-face to give customized and effective tutoring services to guide you succeed. Call us today to schedule a tutoring session and take your math abilities to the next level.
