Hypergeometric Distribution

Hypergeometric distribution

k balls # taken without replacement from an urn with

m white balls and n black balls # (all in all- m+n)

x or q (from the interval [0 - k] ) #is the number of white balls in the sample

               m        n       m+n

P( x, k, m, n) = ( ) * ( )/ ( ) x k-x k

require('ggplot2')
m <- 5 ; n <- 5
k <- 10
x <- c(0:10)
y <- dhyper(x, m, n, k, log = FALSE)
qplot(x, y, color = I("red"), 
      size = I(2), alpha = I(1/2), geom = c("point"))

Which is totally logical. We drew ALL the balls, 5 of them are white, 5 - black, the probability of 5 white balls is = 1

Let’s eliminate the effect of removal by increasing the overall number of balls in the urn but keep the proportion the same

m <- 1000 ; n <- 1000
k <- 10
y <- dhyper(x, m, n, k, log = FALSE)
qplot(x, y, color = I("green"), 
      size = I(2), alpha = I(1/2), geom = c("point"))

Exactly the same as the Binomial distribution with size = 10 and p=0.5 .

Now let’s dampen the effect of removal gradually

m <- 6 ; n <- 6
k <- 10
y <- dhyper(x, m, n, k, log = FALSE)
qplot(x, y, color = I("steelblue"), 
      size = I(2), alpha = I(1/2), geom = c("point"))

only the neighbours of the central point are nonzero

m <- 8 ; n <- 8
k <- 10
y <- dhyper(x, m, n, k, log = FALSE)
qplot(x, y, color = I("blue"), 
      size = I(2), alpha = I(1/2), geom = c("point"))

On and on it goes…

m <- 10 ; n <- 10
k <- 10
y <- dhyper(x, m, n, k, log = FALSE)
qplot(x, y, color = I("green"), 
      size = I(2), alpha = I(1/2), geom = c("point"))

10 white and 10 black, sample 10 still much sharper than the Binomial with size 10 and p=0.5

Now let’s move it to the sides in the case when the effect of removal is still visible

m <- 15 ; n <- 5
k <- 10
y <- dhyper(x, m, n, k, log = FALSE)
qplot(x, y, color = I("red"), 
      size = I(2), alpha = I(1/2), geom = c("point"))

m <- 5 ; n <- 15
k <- 10
y <- dhyper(x, m, n, k, log = FALSE)
qplot(x, y, color = I("magenta"), 
      size = I(2), alpha = I(1/2), geom = c("point"))

Hypergeometric is just MUCH sharper if the numbers are comparable.

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