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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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