1. Each Distribution for Random Variables Has:
Definition
Parameters
Density or Mass function
Cumulative function
Range of valid values
Mean and Variance

2. Binomial Distribution
Bernoulli trial
S = {success, failure} = {0, 1}
trials are independent P(success) = constant = p
(independent is similar to sampling with replacement)
Binomial Distribution
n = Bernoulli trials with p (1-p = q)
xi = number of success in n trials ~ b(n, p)
xi = 0, 1, …., n
V(X) = np(1-p)
IME 312

3. Hypergeometric Distribution n = number of items in sample taken
(This is similar to Binomial, but sampling without replacement)
n = number of items in sample taken
N = number of items in sample space
r = number of success in sample space of N
xi = number of success in n sample taken ~ h(N, n, r)
Relation to Binomial Distribution
IME 312

4. Geometric Distribution
p = P(success of independent Bernoulli trials) = constant
xi = number of trials up to and including the first success
for xi = 1, 2, 3, 4, …….
V(X) = (1-p)/p2
IME 312

5. Negative Binomial Distribution
p = P(success of independent Bernoulli trials) = constant
xi = number of trials up to and including the r’th success
for xi = r’, r’+1, r’+2, r’+3, r’+4, …….
V(X) = r’(1-p)/p2
IME 312, updated Oct 2012

6. Multinomial Distribution n = number of independent trials
k = possible types of outcome for each trial
xi = number of outcomes from type i i=1 to k
pi = constant probability of having type i outcome
So that x1+x2+ …. +xk=n and p1+p2+….+pk=1
IME 312, updated Oct 2012

7. xi = 0, 1, 2, …... Poisson Distribution P(x > 1 in subinterval) = 0
P(x = 1 in subinterval) = fix and proportional to the length
Independent count in each subinterval
= mean number of counts in the unit interval > 0
x = number of counts in the unit interval
Unit Matching between x and !
Approximation of Binomial by Poisson:
xi = 0, 1, 2, …...
IME 312