
260 Combinatorics of Permutations, Second Edition
f
F
Normal Density Normal Distributio
FIGURE 6.12
Normal density and normal distribution.
and σ =1,andZ has normal distribution, then we say that Z has standard
normal distribution.
Let X
n
be a random variable, and let a
n
(k) be a triangular array of non-
negative real numbers, n =1, 2, ···,and0≤ k ≤ m(n)sothat
P [X
n
= k]=p
n
(k)=
a
n
(k)
m(n)
i=0
a
n
(i)
.
Set g
n
(x)=
m(n)
k=0
p
n
(k)x
k
.Inotherwords,g
n
(x) is a generating function
satisfying g
n
(1) = 1. Such generating functions are sometimes called proba-
bility generating functions.
Let us introduce the following notation that is standard in most probability
texts.
• Let
¯