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,,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,

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

XI=outcomes of the ith kind.

Multivariate Hypergeometric

N individuals in the population

k possible outcomes

Mi=number of kind i in the population

n individuals are selected without replacement

XI=outcomes of the ith kind.