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Discrete Distributions

Distributions Summary

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Name Parameters p.d.f Expectation Variance p.g.f
Binomial
B ( ~n , ~p )
~n &in. &naturals.
~p &in. ( 0 , 1 )
( ^~n_~x ) ~p^~x ( 1 - ~p )^{~n - ~x} ~n~p ~n~p ( 1 - ~p ) ( ~p~t + ( 1 - ~p ) )^~n
Negative Binomial ~r &in. &reals.^+
~p &in. ( 0 , 1 )
( ^{~r +}_~x^{~x - 1} ) ~p^{~r} ( 1 - ~p )^~x ~r ( 1 - ~p ) ./ ~p ~r ( 1 - ~p ) ./ ~p^2 ~p^~r ./ ( 1 - ( 1 - ~p )~t ) ^~r
Geometric
G_0( ~p )
~p &in. ( 0 , 1 ) ~p ( 1 - ~p )^~x ( 1 - ~p ) ./ ~p ( 1 - ~p ) ./ ~p^2 ~p ./ ( 1 - ( 1 - ~p )~t )
Hypergeometric ~r, ~n, ~s &in. &naturals.
1 < ~s < r
1 < ~n < ~r
~p &in. ( 0 , 1 )
( ^~s_~x )( ^~r_~n^{-}_{-}^~s_~x ) ./ ( ^~r_~n ) _ _ _
Poisson
Poisson(&mu.)
&mu. &in. &reals.^+ &mu.^~x e^{-&mu.} ./ ~x#! &mu. &mu. e^{&mu. ( ~t - 1 )}

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