Download STOCHASTIC PROCESSES - RANDOM VARIABLES

3 RANDOM VARIABLES
Random variable is a function that maps the sample space S into
the extended real line.
We denote the real line as 
 (- < x < +)
and the extended real line as  + =
Formal definition:
S
{}
X : S  +
P(  S : X( ) = ) = 0

X( )
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+
Note: Random variable is not the variable in the usual
sense, but function.
Two sample points might be assigned to the same value
of X, i. e. X(1)
= X(2), but one sample point can not be
assigned to two different values of X.
S
2
1


X( )
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S
+
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
X1 ( )

X2 ( )
+
The sample space S is called the
domain of the random variable X.
Collection of all the values of X is called the
range of the random variable X.
Example 3-1: In the experiment of tossing a coin once we might
define the random variable as:
X(H) = 0, X(T) = 1
or
X(H) = 10, X(T) = 15
S
S
H
H
T
0
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1
+
3-3
10
T
15
+
Example 3-2: In the fair die experiment, we assign to the six
outcomes f1, f2, …, f6, the numbers X(fi) =10i. Thus, we have:
X(f1) = 10, X(f2) = 20, X(f3) = 30,…., X(f6) = 60,
S
f1
10
20
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f2
30
f3
f4
f5
40
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f6
50
60
Events Defined by Random Variables
If X is a random variable, and x is a fixed real number, we can
define the event (X
= x) as:
X  x   : X    x
Similarly, for fixed numbers x, x1, and x2, we can define the
following events:
X  x   : X    x
X  x   : X    x
x1  X  x2    : x1  X    x2 
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We can ask ourselves what are the probability of these events.
Probabilities are defined by:
P X  x  P : X    x
P X  x  P : X    x
P X  x  P : X    x
Px1  X  x2   P : x1  X    x2 
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Example 3-3:
In the experiment of tossing a fair coin three times, the sample
space S consists of eight equally likely sample points:
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
If X is a random variable giving the number of heads obtained,
find:
(a) P(X
= 2); (b) P(X < 2).
(a) Let A
S be the event defined by X = 2:
(b) Let B
S be the event defined by X < 2:
A   X  2   : X    2  HHT , HTH ,THH 
P X  2  P A  3 / 8
B   X  2   : X    2  HTT ,THT ,TTH ,TTT 
P X  2  PB  4 / 8  1/ 2
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Distribution function
The distribution function (or cumulative distribution function of X is
the function defined by:
FX x   P X  x     x  
Example 3-4: In the experiment of tossing a coin (not fair) once,
we defined the random variable as
probabilities of the of the events
X(H) = 0, X(T) = 1, with
P X  0  p; P X  1  q  1  p
Find the distribution function.
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x
( X  x)
PX (x)
- < x <0
0
0 x <1
1
1  x < +

{H}
{H}
{H,P}
{H,P}
0
p
p
p+q=1
1
FX x
1
p
-1
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0
+1
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x
Properties of the distribution function FX (x):
1.
0  FX ( x)  1
2.
FX ( x1 )  FX ( x2 )
3.
lim FX ( x)  FX ()  1
4.
lim FX ( x)  FX ()  0
5.
if x1  x2
x 
x  
lim FX ( x)  FX (a  )  FX (a)
x a
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Determination of the Probabilities from the Distribution function:
Pa  X  b  FX b  FX a
P X  a  1  FX a

P X  b  FX b 
P X  x0   FX ( x0 )  FX x0 
FX (x)
1
FX ( x0 )
x0
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x
For the continuous random variable:
P X  x0   0
For the discrete random variable:
P X  xi   FX ( xi )  FX xi 1 
 P( X  xi )  P( X  xi 1 )
P X  x  pX (x) ,for discrete random variable,
is called the
probability mass function.
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Probability density function
The derivative
dFX ( x)
f X ( x) 
dx
is called the probability density function of the continuous
random variable X.
If FX (x) has a jump discontinuity at the point x0, then the
probability density function contains the term:
F






(
x
)

F
(
x
)

(
x

x
)

F
(
x
)

F
(
x
X
0
X
0
0
X
0
X
0 )  ( x  x0 )
*
*
*
*
x
FX ( x)  P( X  x)   f X ( )d

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Properties of probability density function fX (x):
1.
f X ( x)  0

2.

f X ( x)  1

3.
fX (x) is piecewise continuous
b
4.
P(a  X  b)   f X ( x)dx
a
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Mean value and Variance:
The mean (or expected) value of a random variable X,
denoted by X or E(X), is defined by:
  xk p X ( xk ) X : discrete
 k
 X  E ( X )   
 xf X ( x)dx X : continouos
 
The nth moment of a random variable X is defined by:
  xkn p X ( xk )
 k
n
mn  E ( X )    
 x n f X ( x)dx
 

X : discrete
X : continouos
The nth moment about the mean is defined by
m  E  X   X 
Stochastic Processes – Random Variables
m
3-15

The Variance of a random variable X is defined by:

  Var ( X )  E  X   X 
2
X
2

  xk   X 2 p X ( xk ) X : discrete
 k
2
 X   
2
 x   X  f X ( x)dx X : continouos
 
  E( X )  
2
X
2
2
X
The standard deviation X of a random variable X is defined by:
X   
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2
X
SOME SPECIAL DISTRIBUTIONS
Uniform distribution:
 1

f X ( x)   b  a
0

fX (x)
a xb
otherwise
1
ba
a
 0
x  a
FX ( x)  
b  a
 1
xa
a xb
xb
FX (x)
1
a
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b
x
b
x
Poisson Distribution
p X (k )  P( X  k )  e


k
k!
pX (x)
x
FX ( x)  e

n

k 0
X  
Stochastic Processes – Random Variables

k
k!
n  x  n 1
 
2
X
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Normal (or Gaussian) Distribution
1
f X ( x) 
e
2 
1
FX ( x) 
2 
X  
By taking
 X  0 and 

x   2


2 2
x


   2

e
2 2
d
 
2
X
2
X
2
 1 we get the standard
normal distribution
.
You can se the diagram of the normal distribution by going to:
http://playfair.stanford.edu/~naras/jsm/NormalDensity/NormalDen
sity.html
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By introducing the change of variable
y  (   X ) / 
and the function  (z)
1
( z ) 
2

z

e
y2

2
dy;
( z )  1   ( z )
we can express the normal distribution as:
x X
1
FX ( x) 
2



Error function is defined by:
Stochastic Processes – Random Variables
 x  X 
e dy  

  
2 z t 2
erf ( z ) 
e

y2

2

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0
Conditional distributions:
The conditional distribution function FX
(x|B) of the random
variable X, under the condition that event B happens first, is
given by:
P( X  x)  B 
FX ( x | B)  P( X  x | B) 
P( B)
It has the same properties as FX
Also:
(x)
P( X  xk )  B 
p X ( xk | B)  P( X  xk | B) 
P( B)
dFX ( x | B)
f X ( x | B) 
dx
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