Hypergeometric distribution

In a given population, we partition it into two groups $A$ and $B$, where $a$ and $b$ are the number of people in each group. We then sample $n$ people without replacement.

Let $X$ be the Random variable that equals the number of people from group $A$. Then $X$ is said to have a Hypergeometric distribution with parameters $a$, $b$, and $n$. Textbook writes $X\sim \text{HGeom}(a,b,n)$.

Note that we are sampling without replacement to get our group of $n$ people.

What distribution would X have if we sampled with replacement?

The Binomial distribution.

We can show this by considering that the number of samples is $n$, and the success rate (picking a person from group $A$) is $\frac{a}{a+b}$. In addition, each of the $n$ trials are independent Bernoulli trials.

Then we can say $X\sim \text{Bin}\left(n,\frac{a}{a+b}\right)$.

PMF

Each sample size of $n$ is a valid outcome in our experiment. We can count the number of outcomes, which is given by $\left(\genfrac{}{}{0ex}{}{a+b}{n}\right)$. They’re all equally likely.

To find $P(X=k)$, we need to count the number of ways to choose exactly $k$ people from group $A$ out of the $n$ sample size.

• $\left(\genfrac{}{}{0ex}{}{a}{k}\right)$ ways to choose $k$ people from group $A$.
• $\left(\genfrac{}{}{0ex}{}{b}{n-k}\right)$ ways to choose the remaining $n-k$ people from group $B$.

Therefore, we have

$$P(X=k)=\frac{\left(\genfrac{}{}{0ex}{}{a}{k}\right)\left(\genfrac{}{}{0ex}{}{b}{n-k}\right)}{\left(\genfrac{}{}{0ex}{}{a+b}{n}\right)}$$

for integers $0\le k\le w$ and $0\le n-k\le b$.

Expectation

We can use Indicator variables for Hypergeometric distributions as well. Given $X\sim \text{HGeom}(a,b,n)$, we let $X$ be the sum of $n$ indicator variables, where $I_i=1$ if the $i$th person is included in the sample.

$$\begin{array}{rl}X& =\sum _{i=1}^n{I}_i\\ \mathbb{E}(X)& =\sum _{i=1}^n\mathbb{E}(I_i)=n\cdot \left(\frac{a}{a+b}\right)=\frac{na}{a+b}\end{array}$$

This is assuming that $X$ equals the number of people from group $A$ in our sample of $n$. Without loss of generality this applies for counting $b$ as well.