\defl{Hypergeometric distribution has the following properties:}
- When units are selected from a finite population without replacement and the population consists of successes and failures.
- The major difference between the Hypergeometric distribution and the Binomial distribution is that the probability of selecting a success is {\bf not constant and is not independent} from each draw.
\defm{Hypergeometric Distribution}
\[ P(X=x)=f(x) = \frac{{A\choose x}{{N-A}\choose {n-x}}}{{N\choose n}} \]
- $X$ has a Hypergeometric distribution.
- $x$ is the number of successes in the sample
- $n$ is the sample size
- $A$ is the number of successes in the population
- $N$ is the population size
- $N-A$ is the number of failures in the population
- $n-x$ is the number of failures in the sample
- $\mu = E(X) = \frac{nA}{N}$
- $\sigma^2 = V(X) = \left(\frac{N-n}{N-1}\right) \frac{nA(N-A)}{N^{2}}$
- $\frac{N-n}{N-1}$ is called the finite population correction factor.
\defs{Examples}
- The number of spades selected when 5 cards are drawn from a standard 52 deck of cards.
- There are 20 Sony CD players in stock at the Sony store at the Mall Bangkapi; 5 are defective. A customer buys 6 of the 20 CD players. The number of defective CD players bought of the 6 CD players.
- The computer lab has 20 computers and 15 of the 20 computers have illegal software on them. The number of computers selected with illegal software from a sample of size 5.