This applet computes probabilities for the hypergeometric distribution
$$X \sim HG(n, N, M)$$
where
- $n = $ sample size
- $N = $ total number of objects
- $M = $ number of successes (note: number of failures is $N-M$)
The random variable $X$ equals the number of successes in the sample. The support of $X$ takes the usual restrictions:
- $0 \leq x \leq n$
- $x \leq M$
- $n-x \leq N-M$
Directions
- Enter the sample size in the $n$ box.
- Enter the total number of objects the $N$ box.
- Enter the number of successes in the $M$ box.
- Hitting "Tab" or "Enter" on your keyboard will plot the probability mass function (pmf).
To compute a probability, select $P(X=x)$ from the drop-down box,
enter a numeric $x$ value, and press "Tab" or "Enter" on your keyboard. The probability
$P(X=x)$ will appear in the
pink box.
Select $P(X \leq x)$ from the drop-down box for a left-tail probability (this is the cdf).
Details
-
$f(x)=P(X=x)=\frac{{M \choose x}{N-M \choose n-x}}{N \choose n}$
for $x=0,1,\ldots,n$ where $x \le M$ and $n-x \le N-M$
- $\mu=E(X)=n\frac{M}{N}$
- $\sigma^2=Var(X)=n\frac{M}{N}\left(1-\frac{M}{N}\right)\left(\frac{N-n}{N-1}\right)$
- $\sigma=SD(X)=\sqrt{n\frac{M}{N}\left(1-\frac{M}{N}\right)\left(\frac{N-n}{N-1}\right)}$