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GENERATING FUNCTIONS:
Definition:Let {a0 , a1 , a2 ,
...
If
2

A(s) = a0 + a1 s + a2 s +
...

Example
Let ak =

1
,k
k!

= 0, 1, 2,
...
= es
0!
1!
2!
3!

A generating function is a convergent power series which can be finite of infinite
...

PROBABILITY GENERATING FUNCTION(p
...
f ):
We consider two definitions
Definition 1
This definition looks at probability generating function (pgf) as a special case of a generating
function
...
The corresponding A(s) is called
a probability generating function(p
...
f)
...


1

Instead of ak we usually use pk , the probability that a random variable X takes a non-negative
integer
...
g
...

Definition 2
The generating function G(s) of the integer-valued random X is defined by
X

G(s) = E[S ] =

∞
X

P rob(X = k)sk

k=0

Deriving moments of random variable using p
...
f
From G(s) = E[S X ],the first and second derivatives of G(s) with respect to s are
dG(s)
= G′ (s) = E[XS X−1 ]
ds
d2 G(s)
= G′′ (s) = E[X(X − 1)S X−2 ]
ds2
Substituting s = 1, we have
G(1) =

X

pk = 1

G′ (1) = E[X]
G′′ (1) = E[X(X − 1)]
the mean and variance of X are
E[X] = G′ (1)
and
V ar(X) = E(X 2 ) − [E(X)]2
= E[X(X − 1)] + E[X] − [E(X)]2
= G′′ (1) + G′ (1) − [G′ (1)]2
Examples of p
...
f:
(i) Let X have binomial distribution with parameters n and p;
 
n x
P (X = x) =
p (1 − p)n−x , x = 0, 1,
...
g
...

Hence the mean and variance of X are
E[X] = G′ (1) = np
and
V ar(X) = G′′ (1) + G′ (1) − [G′ (1)]2
= n(n − 1)p2 + np − (np)2
= np(1 − p)
(ii) Let X have Poisson distribution with parameter λ;
P (X = x) =

e−λ λx
,
x!

x = 0, 1, 2,
...
, Xp )′ be a p-dimensional random vector with joint probability distribution
function f (x1 , x2 , x3 ,
...
The joint probability generating function of the vector is defined by


G(s) = E S1X1 S2X2
...

sx1 1 sx2 2
...
, xp )

xp

if the expectation exists for all values s = [s1 , s2 ,
...
g
...
, p can be obtained from the joint p
...
f by substituting
sj = 1;

sj , j ̸= i ;i
...
si , 1, 1
...
g
...

G(s1 , s2 , s3 ,
...
, sk , 1, 1
...
g
...
, p equal to one

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