Download Uniform Random Variant

Exponential Random Variable
A continuous random variable X is called exponentially distributed with the parameter   0
if the probability density function is of the form
x0
 e   x
f X ( x)  
otherwise
 0
The corresponding probabilty distribution function is
x
FX ( x) 

f X (u )du

x0
0

 x
x0
1  e

We have  X  EX   x e   x dx 
0

1
1

1
 X2  E ( X   X ) 2   ( x  ) 2  e   x dx  2


0
The following figure shows the pdf of an exponential RV.



The time between two consecutive occurrences of independent events (Waiting time )can be
modeled by the exponential RV. For example, the exponential distribution gives probability
density function of the time between two successive counts in Poisson distribution
Used to model the service time in a queuing system.
In reliability studies, the expected life-time of a part, the average time between successive
failures of a system etc., are determined using the exponential distribution.
Example
The waiting time of packets in X in a computer network is an exponential RV with
f X ( x)  0.5e 0.5 x
x0
0.5
P ({0.1  X  0.5}) 
 0.5e
0.5 x
dx
0.1
=e 0.50.5  e 0.50.1
 0.0241
Memory less property of the exponential Distribution
For an exponential RV X with parameter  ,
P ( X  t  t0 / X  t0 )  P ( X  t ) for t  0, t0  0
Proof:
P[( X  t  t0 )  P  X  t0 ]
P ( X  t0 )
P ( X  t  t0 / X  t0 ) 

P ( X  t  t0 )
P ( X  t0 )

1  FX (t  t0 )
1  FX (t0 )
e   ( t  t0 )
e   t0
 e   t  P( X  t )
Hence if X represents the life of a component in hours, the probability that the component will last
more than t  t0 hours given that it has lasted t 0 hours, is same as the probability that the component
will last t hours. The information that the component has already lasted for t 0 hours is not used. Thus
the life expectancy of a used component is same as that for a new component. Such a model cannot
represent a real-world situation, but used for its simplicity.

Laplace Distribution
A continuous random variable X is called Laplace distributed with the parameter   0 with the
probability density function is of the form
f X ( x) 

2
e
 x
  0, -  x  

We have  X  EX 

 x2e
 x
dx  0


  E( X   X )   x2
2
X
2


2
e
 x
dx 
2
2
Chi-square random variable
A random variable is called a Chi-square random variable with n degrees of freedom if its
PDF is given by
2

x n / 21
e x / 2 x 0
 n/2 n
f X ( x)   2  (n / 2)
0
x0

with the parameter   0 and (.) denoting the gamma function. A Chi-square random
variable with n degrees of freedom is denoted by  n2 .
Note that a  22 RV is the exponential RV.
The pdf of  n2 RVs with different degrees of freedom is shown in Fig. below:
Mean and variance of the chi-square random variable

 X   xf X ( x)dx


x
0
2
x n / 21
e  x / 2 dx
n/2 n
2  (n / 2)

2
xn / 2
e  x / 2 dx
n/2 n
 (n / 2)
0 2

n
(2 2 ) n / 2 u 2  u
  n/2 n
e (2 2 )du ( Substituting u  x / 2 2 )
 (n / 2)
0 2


2 2 [(n  2) / 2]
(n / 2)

2 2 n / 2(n / 2)
(n / 2)
 n 2
Similarly,

EX 2   x 2 f X ( x )dx


  x2
0
2
x n / 21
e  x / 2 dx
n/2 n
2  (n / 2)
2
x ( n  2) / 2
e  x / 2 dx
n/2 n
 (n / 2)
0 2


( n 2 ) / 2
(2 2 )( n  2) / 2 u
  n/2 n
e  u (2 2 )du ( Substituting u  x / 2 2 )
2  (n / 2)
0



4 4 [( n  4) / 2]
(n / 2)
4 4 [(n  2) / 2]  n / 2  ( n / 2)
(n / 2)
 n(n  2) 4
 X2  EX 2   X2  n(n  2) 4  n 2 4  2n 4
A random variable Let X 1 , X 2 . . . X n
be independent zero-mean Gaussian variables each with
variance  X2 . Then Y  X 12  X 22  . . .  X n2 has  n2 distribution with mean n 2 and variance 2n 4 .
This result gives an application of the chi-square random variable.
Relation between the Chi-square distribution and the Gaussian distribution
A random variable Let X 1 , X 2 . . . X n
be independent zero-mean Gaussian variables each with
variance  X2 . Then Y  X 12  X 22  . . .  X n2 has  n2
distribution with mean n 2 and variance
2n  4
Rayleigh Random Variable
A Rayleigh random variable is characterized by the PDF
 x  x 2 / 2 2
x 0
 e
f X ( x )   2
0
x0
where  is the parameter of the random variable.
The probability density functions for the Rayleigh RVs are illustrated in Fig.
Mean and variance of the Rayleigh distribution

EX 
 xf
( x)dx
X


x
x

0

2

e x
2
/ 2 2
x2


2
2 
0
dx
e x
2
/ 2 2
dx
2  2
 2



2

Similarly,

EX 2 
x
2
f X ( x)dx


  x2
0
 2
x

2
e x
2
/ 2 2
dx

2
u
 ue du
( Substituting u 
0

 2 2
( Noting that
 ue
u
x2
)
2 2
du is the mean of the exponential RV with  =1)
0
 X2  2 2  (
 (2 

2

2
 )2
) 2
Relation between the Rayleigh distribution and the Gaussian distribution
A Rayleigh RV is related to Gaussian RVs as follow:
If X 1 ~ N (0,  2 ) and X 2 ~ N (0,  2 ) are independent, then the envelope X  X12  X 22 has the
Rayleigh distribution with the parameter  .
We shall prove this result in a later lecture. This important result also suggests the cases where the
Rayleigh RV can be used.
Application of the Rayleigh RV
 Modelling the root mean square error Modelling the envelope of a signal with two orthogonal components as in the case of a signal
of the following form:
s(t )  X1 cos wt  Y1 sin wt
Gamma random variables
A
random
variable
X
is
called
a
gamma
random
variable
with
parameters
  0 and   0 freedom if its PDF is given by
   x 1  x / 
e
x 0

f X ( x)   ( )
0
x0

where (.) is the gamma function.  and  are called shape and scale parameters respectively.

The gamma distribution can assume different shapes for different values of  and  as shown
in Fig. below. For   1, f X ( x) is a strictly decreasing function of x asymptotically
approaching 0 as n . For   1, f X ( x) has a unique mode at the location
 1
x

and of value

(  1) 1  ( 1)
e
( )
The gamma distribution is important because many practical data can be fitted with it.

Many distributions may be considered as special cases of the gamma distribution.   1 gives
k
the exponential distribution.. With   2 and   for positive integer k , we get the chi2
square distribution.

The Erlang distribution is a special case of the gamma distribution where the shape parameter
1
 is an integer. With   k and   ,   0, the Erlang PDF is given by

  k x k 1   x
e
x0

f X ( x)   (k  1)!
0
x0

 If X1 , X 2 ,..., X k are iid exp( ) random variables, then Y  X1  X 2  ...  X k has Erlang
distribution with   k.
Mean and variances of the Gamma random variable
The mean of the Gamma random variable X is given by

EX   x
0
  x 1  x / 
e dx
( )
 u u
e du
0  ( )
(  1)

( )
 
Similarly

  x 1  x / 
EX 2   x 2
e dx
0
( )
 2 (  2)

( )
2
  (  1)
 X2  EX 2  ( EX ) 2   2


( substituting u  x /  )