Download List of Discrete Distributions

PROBABILITY: The Mathematical Analysis of Chance Processes.
Definitions and Notation:
W = sample space or outcome space.
An event is a subset of W .
A random variable is a function \ À W Ä dÞ
Chance processes are described and analyzed mathematically using random
variables.
Discrete Random Variables:
\ À W Ä d is a discrete random variable if \ assumes at most countable
many values. (The values of X can be listed in a sequence.)
:X (x) = T ( \ = B ) is called the probability frequency function or
probability mass function for X.
J\ (B) = T ( X Ÿ B ) is called the distribution function or
cumulative distribution for \ .
Note:
lim J (B) = 0 , and
BÄ -_
lim J (B) = 1 .
BÄ _
J is non decreasing and right continuousÞ
Expected value of \ = IÐ\Ñ = ! Bk T ( \ = Bk ) = ! Bk :\ (B) œ .\
k
k
Note: Expected value of g(\ ) = ! g(Bk ) :\ (B)
k
2
2
Variance of \ = I (\  .\ )2 = E(\ 2 ) - .\
= 5\
.
Standard Deviation of \ = 5\ .
Some Important Discrete Random Variables:
1. Geometric:
\ = number of independent trials until the first success where
: = probability of success on any given trial.
p(k) = P(X=k) = (1-p)k-1 p = qk-1 p , k = 1, 2, 3, ...
1
p
E(X) =
, var(X) =
q
p2
2. Binomial:
X = number of successes in n independent trials where
p = probability of success on any given trial.
p(k) = P(X=k) = a nk bpk (1-p)n-k , k = 0, 1, 2, ... , n
E(X) = np , var(X) = npq
3. Negative Binomial:
X = number of independent trials until r successes where
p = probability of success on any given trial.
‰ r k-r
p(k) = ˆ k-1
r-1 p q
E(X) =
r
p
,
, k = r , r+1, r+2, ...
var(X) =
r (1-p)
p2
4. Hypergeometric:
Draw n balls, without replacement, from a box containing N balls, of
which m are white and N-m are black. Let X = number of white
balls selected.
p(k) =
E(X) =
7 ‰
ˆ7
‰ˆ R85
5
R
ˆ8‰
nm
N
ß 5 œ !ß "ß #ß ÞÞÞ ß 8
, var (X) =
N-n
N-1
np(1-p)
Note: In statistics, the hypergeometric distribution is called "sampling
without replacement".
5. Poisson:
X = the number of rare events occurring in any fixed interval or region..
p(k) = e-
-k
k!
, k = 0, 1, 2, ....
E(X) = - , Var (X) = -
Note: For n large and p small , the binomial distribution is
approximately a Poisson distribution with - = np .