Hypergeometric Distribution in Business Statistics : Meaning, Examples & Uses

What is Hypergeometric Distribution?

Hypergeometric Distribution is a discrete probability distribution that models the probability of drawing a particular number of successes (usually classified as ‘x’) without an alternative from a finite population of items, containing each success and failure. In simpler terms, it is used to calculate the probability of obtaining a certain number of successful outcomes from a finite population while every draw affects the probability of the next draw. This makes it particularly valuable in scenarios where the population size is small, and sampling is finished without replacement.

The essential parameters of the Hypergeometric Distribution are:

• N: The general population size.
• k: The number of success consequences within the population.
• n: The number of draws or samples without replacement.
• X: The number of successful results inside the sample.

Probability Density Function (PDF)

Probability density function (PDF) for the Hypergeometric Distribution is a mathematical function that describes the probability of observing a specific value ‘x’ (the number of successes) in a sample of size ‘n’ drawn from a finite population of size ‘N’ containing ‘k’ successes. It can be described as:

Here,

• P(X = x) represents the probability of obtaining ‘x’ successes.
• is the number of ways to choose ‘x’ successes from ‘k’.
• is the number of ways to choose ‘n – x’ failures from ‘N – k’.
• is the total number of ways to choose ‘n’ items from a population of ‘N’.

Mean and Variance

I. Mean: Mean calculates the average number of successful outcomes you can expect to obtain in a sample of size n, family population of N with k successful outcomes. It is given by the formula,

II. Variance: Variance represents the degree of dispersion or spread of the distribution. A higher variance indicates a wider spread, while a lower variant suggests that the data points are closer to the mean.

Examples of Hypergeometric Distribution

Example 1: Employee Promotion

Consider a company with 100 employees, of which 20 are eligible for promotion. The HR department plans to promote five employees. What is the probability that exactly three out of the five promoted employees are eligible?

Solution:

Using the Hypergeometric Distribution formulation:

• N (general employees) = 100
• k (eligible for promoting) = 20
• n (number of promotions) = 5
• x (number of eligible amongst promoted) = 3

P(X=3) = 0.05

Example 2: Quality Control

A manufacturing facility produces a batch of 25 electronic components, of which 10% are expected to be defective. An inspector selects 8 components for testing. What is the probability that 2 of the 8 components are defective?

Solution:

• N (total additives) = 25
• k (faulty components) = 0.10 * 25 = 2.5 ~ 3
• n (number of components tested) = 8
• x (number of defective components among those tested) = 2

P(X=2) = 0.21

When to Use the Hypergeometric Distribution?

The Hypergeometric Distribution is suitable in situations wherein:

1. Sampling is done without replacement from a finite population.
2. The success or failure of an event impacts the probability of the next events (non-independent occasions).

This distribution is usually used in fields like quality control, finance, epidemiology, and genetics, wherein the population size is limited, and every observation isn’t independent of the other.

Difference between Hypergeometric Distribution and Binomial Distribution

Basis	Hypergeometric Distribution	Binomial Distribution
Population Size	Finite and sample without replacement	Finite or Infinite and With or without replacement
Dependency of Trials	Non-Dependent	Dependent
Formula	Involves combination of Binomial coefficients	Simple Probability formula
Use Cases	Quality control and Population analysis	coin flips and product defects
Parameters	N, k, n and x	n, p

Conclusion

In the realm of business statistics, Hypergeometric Distribution performs an essential position in studying and predicting consequences when dealing with finite populations and non-independent events. By understanding its probability density function, mean, and variance, we gain a precious tool for making informed decisions in various situations. Comparing the Hypergeometric Distribution with the Binomial Distribution highlights their awesome capabilities and packages. While the Hypergeometric Distribution is applicable for non-unbiased, finite population eventualities, the Binomial Distribution is more flexible and applicable in a wide number of conditions.

In conclusion, the Probability Density Function (PDF) is an essential concept in probability theory, providing a way to describe the likelihood of a continuous random variable taking up specific values or falling inside certain intervals. The PDF allows us to understand the shape and characteristics of continuous probability distributions, making it a crucial tool for analysing and modelling real-world data in various fields, consisting of statistics, science and engineering.