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Hypergeometric Distribution

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Hypergeometric Distribution

Suppose we have ~n objects (the "population") of which a number, ~r, have a certain property. If we take a sample of ~s objects, then the probability that ~x of these have the property is

P( ~X = ~x ) _ = _ fract{comb{~s,~x} comb{~r - ~s,~n - ~x},comb{~r,~n}}

This is known as the #~{hypergeometric distribution}.

Note that _ ~r =< ~n , _ ~s =< ~n , _ but ~s is not necessarily =< ~r . _ _ ~x =< min\{ ~s , ~r \}.

Moments

The of the Negative Binomial Distribution is given by

&Phi.( ~t ) _ = _ sum{comb{~s,~x} comb{~r - ~s,~n - ~x} ./ comb{~r,~n},~x = 0,~n} _ = _

&Phi.'( ~t ) _ = _

&Phi.''( ~t ) _ = _

&Phi.'( 1 ) _ = _ , _ _ _ _ &Phi.''( 1 ) _ = _

E( ~X ) _ = _

var( ~X ) _ = _

E( ~X ) _ = _
var( ~X ) _ = _