Download Uniform Random Variant

Uniform Random Variable   a  b  
A continuous random variable X is called uniformly distributed over the interval [a, b],
  a  b   , if its probability density function is given by
 1

f X ( x)   b  a
 0
a xb
otherwise
We use the notation X ~ U (a, b) to denote a random variable X uniformly distributed over the interval

[a,b]. Also note that


b
1
dx  1.
ba
a
f X ( x)dx  
Distribution function FX ( x)
For x  a
FX ( x)  0
For a  x  b
x

f X (u )du

x
du
ba
a

xa
ba
For x  b,

FX ( x )  1
Mean and Variance of a Uniform Random Variable

b
x
 X  EX   xf X ( x)dx  
dx
ba

a

ab
2

EX 
2



b
x2
dx
ba
a
x f X ( x)dx  
2
b 2  ab  a 2
3
b 2  ab  a 2 (a  b) 2
  EX   

3
4
2
(b  a )

12
The characteristic function of the random variable X ~ U (a, b) is given by
b e jwx
 X ( w)  Ee jwx  
dx
aba
e jwb  e jwa

jw  b  a 
Example:
Suppose a random noise voltage X across an electronic circuit is uniformly distributed between -4 V
and 5 V. What is the probability that the noise voltage will lie between 2 v and 3 V? What is the
variance of the voltage?
3
dx
1
P (2  X  3)  
 .
9
2 5  ( 4)
2
X
2
2
X
(5  4) 2 27
 .
12
4
Remark
 The uniform distribution is the simplest continuous distribution
 Used, for example, to to model quantization errors. If a signal is discritized into steps of  ,


and .
then the quantization error is uniformly distributed between
2
2
 The unknown phase of a sinusoid is assumed to be uniformly distributed over [0, 2 ] in many
applications. For example, in studying the noise performance of a communication receiver, the
carrier signal is modeled as
X (t )  A cos( wct  )
where  ~ U (0, 2 ).
 A random variable of arbitrary distribution can be generated with the help of a routine to
generate uniformly distributed random numbers. This follows from the fact that the distribution
function of a random variable is uniformly distributed over [0,1]. (See Example)
Thus if X is a continuous random variable, then FX ( X ) ~ U (0,1).
 X2 
Normal or Gaussian Random Variable
The normal distribution is the most important distribution used to model natural and man made
phenomena. Particularly, when the random variable is the sum of a large number of random variables,
it can be modeled as a normal random variable.
A continuous random variable X is called a normal or a Gaussian random variable with parameters  X
and  X 2 if its probability density function is given by,
f X ( x) 
1
2 X
e
1  xX 
 

2  X 
2
,   x  
where  X and  X  0 are real numbers.
We write that X is N   X ,  X 2  distributed.
If  X  0 and  X 2  1 ,
1  12 x2
f X ( x) 
e
2
and the random variable X is called the standard normal variable.

f X ( x) , is a bell-shaped function, symmetrical about x   X .
 X , determines the spread of the random variable X. If  X 2 is small X is more concentrated
around the mean  X .


FX ( x)  P  X  x 
x
1

2 X
e
1  t X 
 

2  X 
2
dt

Substituting, u 
t  X
X
, we get
xX

1
X
 u2
1
FX ( x) 
 e 2 du
2 
 x  X 
 

 X 
where ( x) is the distribution function of the standard normal variable.
Thus FX ( x) can be computed from tabulated values of  ( x). . The table ( x) was very useful in the
pre-computer days.
In communication engineering, it is customary to work with the Q function defined by,
Q ( x)  1   ( x)

1
2

e

u2
2
du
x
1
Note that Q(0)  , Q( x)  Q( x)
2
If X is N   X ,  X 2  distributed, then
EX   X
var( X )   X 2
Proof:

 xf
EX 
( x)dx 
X
2 X




1
1
X
 X  e
X
 xe
1  xX 
 

2  X 

1
 u2
2
du



udu  X

2 
2

1
 0  X

  u
 X 2

1
e
 X 2

X
2 e
 X 2
u2

2

e
1
 u2
2
2
dx
Substituting,
x  X
u
X
du

du

u2

2
du   X
0

e
Evaluation of

x2
2
dx


Suppose I 
e

x2
2
dx

Then
  x 
I    e 2 dx 


 

2
2
2


e

x2
2

dx  e

y2
2
dy

 


 e

x2  y 2
2
dydx
 
Substituting x  r cos and y  r sin  we get
 
I 
2
e

r2
2
r d dr
0 

 2  e

r2
2
r dr
0

 2  e ds
s
0
 2 1  2
 I  2
(
r2
 s)
2
so that x  u X   X
Var ( X )  E  X   X 



1
2 X
 x   
2 X
2
X
e
1  xX 
 

2  X 
2
dx


1
 2
2
2 2
 X u e
1
 u2
2
 X du
Put

 X 2  2  12 u
 u e du
2 0
2
X2
 2
2
 1
2 t
2  t e dt
x  X
X
 u So that dx   X du
Put
1
t  u 2 so that dt  udu
2
0
X  3
 
 2
 2 1 1
 2 X
 
 2 2
 2
 X  

X2
 2
2
Remark
The gamma function is defined by the integral

x   t x 1e  t dt , x  
0
Following are some important properties of gamma function
x  ( x  1)( x  1)

1
 
2
If x is a positive integer, then x   x 1!