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 \}.
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 ) _ = _ |