Hypergeometric distribution where we have a sample of k balls from an urn containing N, of which m are white and n are black.

expValErl(N = n + m, m, n = N - m, k)

varErl(N = n + m, m, n = N - m, k)

## Arguments

N

Total number of balls (white and black) in the urn. $$N = n + m$$

m

Number of white balls in the urn.

n

Number of black balls in the urn. Can specify n instead of N.

k

Number of balls drawn from the urn, k = 0, 1, ..., m + n.

## Value

Function :

• expValErl gives the expected value.

• varErl gives the variance.

Invalid parameter values will return an error detailing which parameter is problematic.

## Details

The Hypergeometric distribution for $$N$$ total items of which $$m$$ are of one type and $$n$$ of the other and from which $$k$$ items are picked has probability mass function : $$Pr(X = x) = \frac{\left(\frac{m}{k}\right)\left(\frac{n}{k - x}\right)}{\left(\frac{N}{k}\right)}$$ for $$x = 0, 1, \dots, \min(k, m)$$.

## Examples


# With total balls specified
expValErl(N = 5, m = 2, k = 2)
#> [1] 0.8

# With number of each colour of balls specified
expValErl(m = 2, n = 3, k = 2)
#> [1] 0.8

# With total balls specified
varErl(N = 5, m = 2, k = 2)
#> [1] 0.36

# With number of each colour of balls specified
varErl(m = 2, n = 3, k = 2)
#> [1] 0.36