Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is an essential department of math that handles the study of random events. One of the important theories in probability theory is the geometric distribution. The geometric distribution is a distinct probability distribution that models the amount of trials required to get 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 offer examples.
Definition of Geometric Distribution
The geometric distribution is a discrete probability distribution that portrays the amount of trials needed to achieve the first success in a succession of Bernoulli trials. A Bernoulli trial is a test which has two possible 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 applied when the trials are independent, meaning that the consequence of one test doesn’t affect the outcome of the next trial. In addition, the chances of success remains constant throughout all the tests. We can 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 depicts the number of trials needed to achieve the first success, k is the count of trials 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 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 an individual Bernoulli trial.
The mean is the anticipated number of trials needed to obtain the first success. For example, if the probability of success is 0.5, then we expect to obtain the first success following two trials on average.
Examples of Geometric Distribution
Here are handful of primary examples of geometric distribution
Example 1: Tossing a fair coin till the first head shows up.
Let’s assume we flip a fair coin until the initial head shows up. The probability of success (obtaining 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 needed to achieve 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 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 getting 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 achieving the first 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 until the initial six turns up.
Let’s assume we roll an honest die until the first six turns up. The probability of success (obtaining a six) is 1/6, and the probability of failure (achieving all other number) is 5/6. Let X be the irregular variable that depicts the number of die rolls needed to obtain the first 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 initial 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 first six on the third roll is:
P(X = 3) = (5/6)^(3-1) * 1/6 = (5/6)^2 * 1/6
And so on.
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The geometric distribution is a important theory in probability theory. It is utilized to model a wide array of real-life scenario, such as the number of trials needed to obtain the initial success in different situations.
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