Real-time digital signal processing: implementations, ... changes in the input signal is limited by its internal clock rate, so that it may be slow to
Document source : notes.ump.edu.my
FX
1
FX
2
if X
1
X
2
,
3:3:4d
PX
1
< x X
2
FX
2
À FX
1
:
3:3:4e
The probability density function of a random variable x is defined as
f X
dFX
dX
3:3:5
if the derivative exists. Some properties of f(X) are summarized as follows:
f X ! 0 for all X
3:3:6a
PÀI x I
I
ÀI
f XdX 1,
3:3:6b
FX
X
ÀI
f xdx,
3:3:6c
PX
1
< x X
2
FX
2
À FX
1
X
2
X
1
f XdX:
3:3:6d
Note that both F(X ) and f(X ) are non-negative functions. The knowledge of these two
functions completely defines the random variable x.
Example 3.2: Consider a random variable x that has a probability density function
f X 0,
x < X
1
or x > X
2
a,
X
1
x X
2
,
&
which is uniformly distributed between X
1
and X
2
. The constant value a can be
computed by using (3.3.6b). That is,
I
ÀI
f XdX
X
2
X
1
a Á dX aX
2
À X
1
1:
Thus we have
a
1
X
2
À X
1
:
If a random variable x is equally likely to take on any value between the two limits X
1
and X
2
and cannot assume any value outside that range, it is said to be uniformly
distributed in the range [X
1
, X
2
]. As shown in Figure 3.9, a uniform density function is
defined as
INTRODUCTION TO RANDOM VARIABLES
91
Summary :
That is, I ÀI f XdX X 2 X 1 a Á dX aX 2 À X 1 1: Thus we have a 1 X 2 À X 1 : If a random variable x is equally likely to take on any value between the two limits X 1 and X 2 and cannot assume any value outside that range, it is said to be uniformly distributed in the range [X 1 , X 2 ].
Tags :
random,xdx,ariable,function,alue,density,any,between,functions,336b,probability,defined,distributed