A random variable XX
is the assignment of a number - real or complex - to each sample
point in sample space. Thus, a random variable can be
considered a function whose range is a set and whose ranges are,
most commonly, a subset of the real line. The probability
distribution function or cumulative is
defined to be

P
X
x=PrX≤x
P
X
x
X
x

(1)
Note that

XX denotes the random
variable and

xx denotes the
argument of the distribution function. Probability distribution
functions are increasing functions: if

A=ω
Xω≤
x
1
A
ω
X
ω
x
1
and

B=ω
x
1
<Xω≤
x
2
B
ω
x
1
X
ω
x
2
,

(PrA∪B=PrA+PrB)⇒(P
X
x
2
=P
X
x
1
+Pr
x
1
<X≤
x
2
)
A
B
A
B
P
X
x
2
P
X
x
1
x
1
X
x
2
, which means that

P
X
x
2
≥P
X
x
1
P
X
x
2
P
X
x
1
,

x
1
≤
x
2
x
1
x
2
.

The probability density function
p
X
x
p
X
x
is defined to be that function when integrated yields
the distribution function.

P
X
x=∫−∞xp
X
αdα
P
X
x
α
x
p
X
α

(2)
As distribution functions may be discontinuous, we allow density
functions to contain impulses. Furthermore, density functions
must be non-negative since their integrals are increasing.