Discrete Probability Distributions
Name 
Experiment 
Probability Mass Function (pmf), p(x) 
Mean, Variance, Moment Generating Function 
Comments 
Discrete Uniform 
Equally likely k different values 



Bernoulli 
¥ two possible outcomes 
, x=0,1 


Binomial 
¥ two possible outcomes ¥ fixed number of trials (n) ¥ is fixed from
trail to trial ¥ independent trials 
X=the number of
successes out of n trials , x=0,1,É,n 

¥ Let , then 
Negative Binomial 
¥ two possible outcomes ¥ no fixed number of trials ¥ is fixed from
trail to trial ¥ independent trials 
X=the number of
trials at which the kth success occurs. x=k,k+1,É 

¥ If k=1 then it is called a geometric distribution. This
distribution is memoryless. ¥ For the geometric distribution 
Hypergeometric 
¥ N individuals in the population ¥ two possible outcomes M=number of successes in the population ¥ n individuals are selected without replacement 
X=the number of
successes out of n trials 

¥ used when we sample without replacement 
Poisson 
¥ counts number of events in one unit ¥ probability that an event occurs in one unit is same for
all units ¥ the number of events in units are independent 
X=the number of
times an event occurs in one unit 

¥ Poisson Approximation to Binomial If X has Bin(n,p) 
Multinomial 
¥ k possible outcomes ¥ fixed number of trials (n) ¥ is fixed from
trail to trial ¥ independent trials 
X_{I}=outcomes
of the ith kind. 

Multivariate
Hypergeometric 
¥ N individuals in the population ¥ k possible outcomes M_{i}=number
of kind i in the population ¥ n individuals are selected without replacement 
X_{I}=outcomes
of the ith kind. 