If we conduct a series of independent Bernoulli trials , with probability ~p of success on each trial, where we stop on the first success, what is the probability of exactly ~x failures before the first success?
This is exactly the probability of the sequence _ 0, 0, 0, ..., 0, 1 _ , where there are ~x zeroes, i.e
P( ~X = ~x ) _ = _ ~p ( 1 - ~p )^~x , _ _ ~x >= 0
This is called the #~{geometric distribution}. This is a valid probability distribution, i.e.
sum{ ~p ( 1 - ~p )^~x ,0, &infty.} _ = _ 1
From the result for the geometric series.
[ Put ~S _ = _ &sum._0^{&infty.} (1 - p)^~x . _ _ _ (1 - p) ~S _ = _ &sum._1^{&infty.} (1 - p)^~x _ = _ ~S - 1 _ _ => _ _ ~S _ = _ 1/~p ]
The of the Geometric Distribution is given by
&Phi.( ~t ) _ = _ sum{ _ ~p ( 1 - ~p ) ^~x ~t ^~x,~x = 0,&infty.} _ = _ _ ~p ./ ( 1 - ( 1 - ~p )~t
&Phi.'( ~t ) _ = _ ( 1 - ~p ) ~p ( 1 - ( 1 - ~p )~t )^{-2}
&Phi.''( ~t ) _ = _ 2 ( 1 - ~p )^2 ~p ( 1 - ( 1 - ~p )~t )^{-3}
&Phi.'( 1 ) _ = _ ( 1 - ~p ) ./ ~p , _ _ _ _ &Phi.''( 1 ) _ = _ 2 ( 1 - ~p )^2 ./ ~p^2
E( ~X ) _ = _ ( 1 - ~p ) ./ ~p
var( ~X ) _ = _ 2 ( 1 - ~p )^2 ./ ~p^2 _ + _ ( 1 - ~p ) ./ ~p _ - _ ( 1 - ~p )^2 ./ ~p^2
_ _ _ _ _ _ _ _ _ = _ fract{( 1 - ~p )^2 + ~p( 1 - ~p ) , _ ~p^2 _ } _ = _ fract{ 1 - ~p , _ ~p^2 _ }
| E( ~X ) _ = _ ( 1 - ~p ) ./ ~p var( ~X ) _ = _ ( 1 - ~p ) ./ ~p^2 |