Hypergeometric Distribution: Probabilities, Feasible Counts, and Moments

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What is a hypergeometric experiment?

An experiment that samples without replacement from a population consisting of two groups and counts how many sampled items belong to a designated group of interest.

What do the parameters in $X \sim H(r,b,n)$ mean?

$r$ is the size of the group of interest, $b$ is the size of the other group, and $n$ is the sample size. The variable $X$ counts selected items from the group of interest.

How do you calculate $P(X=x)$ for a hypergeometric random variable?

Use $P(X=x)=\frac{\binom{r}{x}\binom{b}{n-x}}{\binom{r+b}{n}}$, which divides the number of samples with exactly $x$ items from the group of interest by the number of all samples of size $n$.

What are the feasible values of $X$?

They range from $\max(0,n-b)$ to $\min(n,r)$, inclusive.

What are the mean and standard deviation formulas?

The mean is $\mu=n\frac{r}{r+b}$. The standard deviation is $\sigma=\sqrt{\frac{rbn(r+b-n)}{(r+b)^2(r+b-1)}}$.

Quiz questions

1. Why are successive selections in a hypergeometric experiment dependent?Answer: Without replacement, each selection changes the composition of the remaining population.
2. In $X \sim H(r,b,n)$, what does $r$ represent?Answer: The size of the group of interest.
3. Which expression gives the probability of exactly $x$ items from the group of interest?Answer: $\frac{\binom{r}{x}\binom{b}{n-x}}{\binom{r+b}{n}}$
4. A sample of 9 students is drawn from 16 business majors and 7 non-business majors. If $X$ counts business majors, what is the smallest possible value of $X$?Answer: 2, since no more than 7 non-business majors can be selected.
5. A committee of 7 is chosen from 15 men and 18 women. If $X$ counts men, which expression represents the probability of more than 4 men?Answer: $P(X=5)+P(X=6)+P(X=7)$ for $X \sim H(15,18,7)$