Distributacalcul (version 0.4.0)

Erl: Hypergeometric Distribution

Description

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

Usage

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

Value

Function :

• expValErl gives the expected value.

• varErl gives the variance.

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

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.

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)

# With number of each colour of balls specified
expValErl(m = 2, n = 3, k = 2)


# With total balls specified
varErl(N = 5, m = 2, k = 2)

# With number of each colour of balls specified
varErl(m = 2, n = 3, k = 2)