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Unformatted text preview: Probability Cheat Sheet Distributions Unifrom Distribution notation U [ a,b ] cdf x- a b- a for x [ a,b ] pdf 1 b- a for x [ a,b ] expectation 1 2 ( a + b ) variance 1 12 ( b- a ) 2 mgf e tb- e ta t ( b- a ) story: all intervals of the same length on the distributions support are equally probable. Gamma Distribution notation Gamma ( k, ) pdf k x k- 1 e- x ( k ) I x> ( k ) = Z x k- 1 e- x dx expectation k variance k 2 mgf (1- t )- k for t < 1 ind. sum n X i =1 X i Gamma n X i =1 k i , ! story: the sum of k independent exponentially distributed random variables, each of which has a mean of (which is equivalent to a rate parameter of - 1 ). Geometric Distribution notation G ( p ) cdf 1- (1- p ) k for k N pmf (1- p ) k- 1 p for k N expectation 1 p variance 1- p p 2 mgf pe t 1- (1- p ) e t story: the number X of Bernoulli trials needed to get one success. Memoryless. Poisson Distribution notation Poisson ( ) cdf e- k X i =0 i i ! pmf k k ! e- for k N expectation variance mgf exp ( ( e t- 1 )) ind. sum n X i =1 X i Poisson n X i =1 i ! story: the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. Normal Distribution notation N ( , 2 ) pdf 1 2 2 e- ( x- ) 2 / ( 2 2 ) expectation variance 2 mgf exp t + 1 2 2 t 2 ind. sum n X i =1 X i N n X i =1 i , n X i =1 2 i !...
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