Important Random Variables
EE570: Stochastic Processes
Dr. Muqaiebl
Based on notes of Pillai
See also
http://www.math.uah.edu/stat
http://mathworld.wolfram.com
Continuous-type random variables
1. Normal (Gaussian): X is said to be normal or Gaussian
r.v, if
f X ( x) 
1
2
2
e ( x   )
2
/ 2 2
(3-29)
.
This is a bell shaped curve, symmetric around the
parameter  , and its distribution function is given by

FX ( x ) 
x

1
2 2
e
 ( y   ) 2 / 2 2
x

dy  G
,
  
(3-30)
where G( x )   1 e dy is often tabulated. Since f X (x)
2
depends on two parameters  and  2 , the notation X N (  , 2 )
will be used to represent (3-29).
f (x)
x
 y2 / 2

X
Thermal noise : Electronics,
Communications Theory

Fig. 3.7
x
2. Uniform: X  U (a, b), a  b, if (Fig. 3.8)
Coding Theory
1


, a  x  b,
f X ( x)   b  a

 0, otherwise.
(3.31)
3. Exponential: X   ( ) if (Fig. 3.9)
Queuing Theory
1

 e  x /  , x  0,
f X ( x)   

 0, otherwise.
1
ba
(3-32)
f X (x)
f X (x)
a
Fig. 3.8
b
x
x
Fig. 3.9
4. Gamma: X  G( ,  ) if (  0,   0) (Fig. 3.10)
 1
 x
x / 

e
, x  0,
f X ( x )   ( )  

0, otherwise.

If   n an integer
Queuing Theory
f X (x )
(3-33)
x
Fig. 3.10
( n )  ( n  1)!.
f X (x )
5. Beta: X   (a, b) if (a  0, b  0) (Fig. 3.11)
0
1
x
1
Fig. 3.11

a 1
b 1
x
(
1

x
)
,
0

x

1
,

f X ( x )    ( a , b)
(3-34)

0,
otherwise.

where the Beta function  ( a , b) is defined as
1
 (a, b)   u a 1 (1  u)b1 du.
0
(3-35)
6. Chi-Square: X   2 (n), if (Fig. 3.12)
1

x n / 21e  x / 2 , x  0,
 n/2
f X ( x )   2 ( n / 2 )
(3-36)

0,
otherwise.

f X (x )
x
Fig. 3.12
Note that  2 (n) is the same as Gamma (n / 2, 2).
7. Rayleigh: X  R( 2 ), if (Fig. 3.13)
2
2
x

 2 e  x / 2 , x  0,
f X ( x )  

 0, otherwise.
f X (x )
(3-37)
x
Fig. 3.13
8. Nakagami – m distribution:
 2  m  m 2 m 1  mx 2 / 
x e
, x0



f X ( x )   ( m )   

0
otherwise

(3-38)
Related to Gaussian, Comm. Theory
9. Cauchy: X C ( ,  ),
f X ( x) 
if (Fig. 3.14)
 /
  (x  )
2
f X (x )
2
,    x  .
10. Laplace: (Fig. 3.15)

(339)
Fig. 3.14
1 |x|/ 
f X ( x) 
e
,    x  .
2
(3-40)
11. Student’s t-distribution with n degrees of freedom (Fig
3.16)
2  ( n 1) / 2

( n  1) / 2 
t 
1  
f (t ) 
,    t  .
T
n ( n / 2) 
f X ( x)
n 
(3-41)
fT ( t )
x
Fig. 3.15
t
Fig. 3.16
x
12. Fisher’s F-distribution
{( m  n ) / 2} m m / 2 n n / 2
z m / 2 1
, z0

(mn) / 2
f z ( z)  
( m / 2) ( n / 2)
( n  mz )

0
otherwise

Other distributions:
Erlang (traffic), Weibull (faiure rate), Poreto ( Economics ,
reliability), Maxwell (Statistical)
The exponential model works well for inter arrival times
(while the Poisson distribution describes the total number of events
in a given period)
(3-42)
Discrete-type random variables
1. Bernoulli: X takes the values (0,1), and
P( X  0)  q,
P( X  1)  p.
(3-43)
2. Binomial: X  B(n, p), if (Fig. 3.17)
 n  k n k
P( X  k )  
, k  0,1,2,, n.
k 
p q
 
(3-44)
3. Poisson: X  P( ) , if (Fig. 3.18)
P( X  k )  e 
k
k!
, k  0,1,2,, .
P( X  k )
P( X  k )
k
12
Fig. 3.17
n
Fig. 3.18
(3-45)
4. Hypergeometric:
P( X  k ) 
m
 
k 
 
 N m 


 n k 


,
N 
 
n 
 
max(0, m  n  N )  k  min( m, n )
(3-46)
5. Geometric: X  g ( p ) if
P( X  k )  pqk , k  0,1,2,, ,
q  1  p.
(3-47)
6. Negative Binomial: X ~ NB ( r, p), if
 k  1 r k  r
P( X  k )  
p q ,

 r 1
k  r, r  1,
.
(3-48)
7. Discrete-Uniform:
P( X  k ) 
1
, k  1,2,, N .
N
(3-49)
We conclude this lecture with a general distribution due