Show that the variance of a hypergeometric (N, D, n)...

Show that the variance of a hypergeometric $(N, D, n)$ distribution is given by expression (3.1.8). Hint: First obtain $E[X(X-1)]$ by proceeding in the same way as the derivation of the mean given in Section 3.1.3.

Key Concepts

Hypergeometric Distribution

This distribution describes the probability of drawing a fixed number of successes from a finite population when sampling is done without replacement. It involves parameters such as the total population size, the number of successes within the population, and the number of draws made, modeling situations where the draws are dependent.

Variance of a Distribution

Variance is a fundamental measure of dispersion that quantifies how much a set of values or a random variable deviates from its mean. In the context of probability distributions, it is computed by taking the expected value of the squared deviations from the mean. For the hypergeometric distribution, the derivation of variance relies on expressions involving the first and second moments.

Expectation and Second Moments

Expectation is the mean or average value of a random variable, while second moments, such as E[X(X-1)], provide information necessary for calculating variance. In many derivations, particularly for discrete distributions, the second moment is used to compute the variance by relating it to both the first moment (mean) and the additional structure of the distribution.

Sampling Without Replacement

This concept refers to drawing items from a population where each item is not returned back to the population after being drawn. This type of sampling introduces dependency between the observations because each draw changes the composition of the remaining population, and it is a key factor in the behavior of the hypergeometric distribution.

Combinatorial Analysis in Probability

Combinatorial methods, such as the use of binomial coefficients, are essential for counting the number of favorable outcomes when deriving moments or probabilities in discrete distributions. In the derivation for the hypergeometric variance, combinatorial techniques are applied to count the ways in which successes can occur in the sample drawn from the population.

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