# Hypergeometric distribution

• Pearce_09
In summary, a hypergeometric distribution is a probability distribution used to model situations where the population size is small and the probability of success changes with each draw. Its parameters include the size of the population, the number of successes, and the number of draws. It differs from a binomial distribution in that it models sampling without replacement. Real-world applications of the hypergeometric distribution include quality control, genetics, and epidemiology. It is related to the binomial distribution as a special case, but the two are fundamentally different in how they model sampling.

#### Pearce_09

Hello,
Im just wondering is there a simply calculated formula for the expected value and variance for the hypergeometric distribution. I know how to do it with long calculations. I know the expected value for the binomial is = np and the variance is = npq = np(1-p) .. Is there something like this for the hyper.. I lost my good stats book so I am not sure..
thanks

Yes, there is a simple formula for calculating the expected value and variance of the hypergeometric distribution. The expected value is given by E[X] = n * (N / N + M), where n is the number of trials, N is the number of success states in the population, and M is the number of failure states in the population. Similarly, the variance is given by Var(X) = n * (N / N + M) * (M / N + M) * (N + M - n) / (N + M - 1). This formula is similar to the one for the binomial distribution, but takes into account the finite population size and the fact that sampling is done without replacement. I hope this helps. Good luck with your calculations!

## 1. What is a hypergeometric distribution?

A hypergeometric distribution is a probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement, where each draw can be either a success or a failure. It is used to model situations where the population size is small and the probability of success changes with each draw.

## 2. What are the parameters of a hypergeometric distribution?

The parameters of a hypergeometric distribution are:

• N: the size of the population
• K: the number of successes in the population
• n: the number of draws

## 3. How is a hypergeometric distribution different from a binomial distribution?

The main difference between a hypergeometric distribution and a binomial distribution is that a hypergeometric distribution models sampling without replacement, while a binomial distribution models sampling with replacement. In other words, in a hypergeometric distribution, the probability of success changes with each draw, whereas in a binomial distribution, the probability of success remains the same for each draw.

## 4. What real-world applications use the hypergeometric distribution?

The hypergeometric distribution has several real-world applications, including:

• Quality control in manufacturing, to determine the probability of finding a certain number of defective items in a batch
• Genetics, to model the probability of inheriting a certain number of genes from a parent
• Epidemiology, to study the spread of diseases in a population

## 5. How is the hypergeometric distribution related to the binomial distribution?

The hypergeometric distribution is a special case of the binomial distribution. When the population size is much larger than the sample size, the hypergeometric distribution approaches the binomial distribution. Additionally, the mean and variance of a hypergeometric distribution can be approximated using the mean and variance of a binomial distribution with certain adjustments. However, the two distributions are fundamentally different in that the hypergeometric distribution models sampling without replacement, while the binomial distribution models sampling with replacement.