Each Distribution for Random Variables Has:

1
Each Distribution for Random Variables Has

• Definition
• Parameters
• Density or Mass function
• Cumulative function
• Range of valid values
• Mean and Variance

2
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
• (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)

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
rth success for xi r, r1,
r2, r3, r4, .
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 i1 to k
• pi constant probability of having type i
  outcome
• So that x1x2 . xkn and p1p2.pk1

IME 312, updated Oct 2012
7
Poisson Distribution P(x gt 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 gt 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