A box contains N1 white balls, N2 black balls, and N3 red balls (N1+N2+N3=N) . A random sample o

step 1 For this exercise, we have the correlation of y sub 1, Y sub 2 is equivalent to n sub 1, n minus

A box contains N1 white balls, N2 black balls, and N3 red...

A box contains $N_{1}$ white balls, $N_{2}$ black balls, and $N_{3}$ red balls $\left(N_{1}+N_{2}+N_{3}=N\right) .$ A random sample of $n$ balls is selected from the box (without replacement). Let $Y_{1}, Y_{2},$ and $Y_{3}$ denote the number of white, black, and red balls, respectively, observed in the sample. Find the correlation coefficient for $\left.Y_{1} \text { and } Y_{2} . \text { (Let } p_{i}=N_{i} / N, \text { for } i=1,2,3 .\right)$

Key Concepts

Multivariate Hypergeometric Distribution

This is the probability distribution that arises when drawing a sample without replacement from a finite population consisting of several distinct categories. It models the counts of items from each category and incorporates the fact that the draws are dependent since the composition of the population changes with each draw. This distribution is key for understanding problems where the total number of items is fixed and the categories are mutually exclusive.

Sampling Without Replacement

Sampling without replacement means that once an item is drawn from the population it is not returned, which introduces dependence among the sampled items. Because the outcome of one draw changes the composition of the population, this method contrasts with sampling with replacement where each draw is independent. This dependency affects both the variance of each count and the covariance between counts from different categories.

Covariance in Finite Population Sampling

Covariance measures how two random variables change together. In the context of sampling without replacement, the counts of items from different categories are negatively correlated because an increase in one count implies a decrease in the available items for other counts. The covariance between counts in a multivariate hypergeometric setting is given by a formula that includes a finite population correction factor, demonstrating the inherent negative dependence.

Correlation Coefficient

The correlation coefficient standardizes the covariance by dividing it by the product of the standard deviations of the variables. In the setting of a multivariate hypergeometric distribution, this coefficient quantifies the degree of negative linear relationship between counts from different categories. Its derivation involves using the established formulas for variance (which also include the finite population factor) and covariance, resulting in an expression that depends only on the proportions of each category in the population.

Finite Population Correction

The finite population correction factor appears in the variance and covariance formulas when sampling without replacement. It accounts for the fact that the population is not infinite, which reduces the variability of the sample counts compared to what would be observed in sampling with replacement. This correction is crucial for accurately capturing the decrease in variability and the induced negative dependence among different category counts.

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