Geometric Distribution - Definition, Formula, Mean, Examples
Probability theory is ac crucial department of math that handles the study of random occurrence. One of the crucial theories 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, derive its formula, discuss its mean, and offer examples.
Explanation of Geometric Distribution
The geometric distribution is a discrete probability distribution which portrays the amount of tests required to accomplish the first success in a series of Bernoulli trials. A Bernoulli trial is a test that has two possible outcomes, usually indicated to as success and failure. For instance, flipping a coin is a Bernoulli trial since it can likewise turn out to be heads (success) or tails (failure).
The geometric distribution is used when the tests are independent, which means that the consequence of one test does not affect the outcome of the next trial. Furthermore, the probability of success remains constant across all the trials. 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 given by the formula:
P(X = k) = (1 - p)^(k-1) * p
Where X is the random variable that portrays the number of trials required to achieve the initial success, k is the number of experiments needed to obtain the initial 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 trials needed to achieve the first 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 expected number of experiments required to achieve the initial success. For instance, if the probability of success is 0.5, therefore we expect to attain the initial success following two trials on average.
Examples of Geometric Distribution
Here are handful of essential examples of geometric distribution
Example 1: Tossing a fair coin until the first head appears.
Let’s assume we toss a fair coin until the initial head appears. 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 count of coin flips needed to get the first head. The PMF of X is provided as:
P(X = k) = (1 - 0.5)^(k-1) * 0.5 = 0.5^(k-1) * 0.5
For k = 1, the probability of achieving 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 obtaining 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 a fair die till the first six appears. The probability of success (getting a six) is 1/6, and the probability of failure (getting all other number) is 5/6. Let X be the random variable which depicts the count of die rolls required 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 getting the first 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 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.
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The geometric distribution is a crucial theory in probability theory. It is used to model a wide array of real-world scenario, for instance the count of tests needed to obtain the first success in several situations.
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