Random Variable
2013
Random Variable
 Two types
 Discrete
 Continuous
Random Variable
 Probability mass function
 Discrete
 P(X = xi) = p(xi)
  p(xi) = 1
Random Variable
 Probability density function
 Continuous
 f(x) = e –x
x>0
 P(X = a) = 0
 -  f(x) dx = 1
 P(a < x < b) = ab f(x) dx
Random Variable
 Expected value
  = E(x)
  =  xi p (xi)
  =  x f(x) dx
Random Variable
 Variance
V ( x )  E[( x   ) 2 ]
 E[ x 2  2 x   2 ]
E

x  ( E ( x))
2
2
x
2
i
p ( xi)  (
2
xi p( xi))
Random Variable
 Standard deviation
SD( X )   
 Sums of R.V.
V (X )
Y  a1 x1  a2 x2
E ( y )  a1 E ( x1 )  a2 E ( x2 )
V (Y )  a1 V ( x1 )  a2 V ( x2 )
2
2
Random Variable
SampleMean  X
SampleVariance  S
2
X


(X  X )


i
n 1
i
n
2

x

2
i
nx
n 1
2
Poisson Probability Distribution
Consider a discrete r.v. which is often useful
when dealing with the number of occurrences
of an event over a specified interval of time.
Suppose we want to find the probability
distribution of the accidents at the intersection
of Rural and Apache during a one week
period.
The R.V. we are interested in is the number of
accidents.
Poisson Probability Distribution
i. The Poisson Distribution provides a good model for the probability
distribution of the number of rare events that occur in space, time,
and volume where  is the average at which events occur.
ii. Define: A r.v. is said to have a Poisson distribution if the p.m.f of
X is
 x e 
P(x) = f(x) =
, x = 0,1, …
x!
where  is the rate per unit time or per unit area
E[ X ]  
iii.
V (X )  
Exponential Distribution
Previously, we discussed the Poisson random variable,
which was the number of events occurring in a given
interval. This number was a discrete r.v. and the
probabilities associated with it could be described by the
Poisson Probability Distribution.
Not only is the number of events a r.v., but the waiting
time between event is also a random variable. This r.v. is a
continuous r.v. for it can assume any positive value.
This r.v. is an exponential r.v. which can be described by
the exponential distribution.
Exponential Distribution
 e
x  0&  0
 x
i. Pdf: f ( x )  
otherwise
 0
where  = rate at which events occur
ii. Correspondingly,
x
F ( x)  P ( X  x)    e x dx  1  e x , x  0
0
E[ X ] 
V (X ) 
1

1
2
iii. An important application of the exponential distribution is to
model the distribution of component lifetime. A reason for its
popularity is because of the “memory-less” property of the
Exponential Distribution
The Uniform Distribution
o The simplest distribution is the one in which a continuous r.v. can assume
any value within a interval [a, b]
Def:
A continuous r.v. X is said to have a uniform distribution on the
interval [a,b] if the probability distribution (pdf) of X is:
 1


a xb
f ( x)   b  a

0
otherwise 
The Uniform Distribution
The cumulative distribution is
x
F ( X )  P( X  x) 
 f ( x)dx

x x
x
a
xa
  f ( x)dx 



ba a ba ba ba

x
1
ba
E[ X ]   xf ( x)dx   x(
)dx 
ba
2


x
(b  a ) 2
V (X ) 
12
x
The Uniform Distribution
Note:
An important uniform distribution is
that for when a = 0 and b = 1, namely
U(0, 1)
A U(0,1) r.v. can be used to simulate
observation of other random variables
of the discrete and continuous type.
The Triangular Distribution
• Continuous Distribution
2( x  a )
f ( x) 
a xb
(b  a )( c  a )
2( c  x )

bxc
(c  b)(c  a )
0
elsewhere
The Triangular Distribution
F ( x)  0
xa
( x  a) 2
F ( x) 
(b  a )(c  a )
(c  x ) 2
1 
(c  b)( c  a )
1
xc
a xb
bxc
The Triangular Distribution
F ( x)  0
xa
abc
E ( x) 
3
a 2  b 2  c 2  ab  ac  bc
V ( x) 
18
Normal Distribution
It is a fact that measurements on many random variables will follow a bellshaped distribution.
Random variable of this type are closely approximated by a Normal
Probability Distribution.
A continuous r.v. X is said to have a normal distribution if the pdf of X is
f ( x) 
1
2 

e
( x )2
2 2
,   0,  x  ,    
The distribution contains 2 parameters ( and ). These are the expected
value and the variance and hence locate the center of the distribution and
measure its spread.
Normal Distribution
The Standard Normal Distribution
To compute P(a  x b) when X ~ N(, 2), we must evaluate
b
b
 f ( x)dx  
a
a
1
2 

e
( x )2
2 2
dx
Note: None of the standard integration techniques can be used
to evaluate this pdf. Instead, for  = 0, and 2 = 1, the pdf has
been evaluated and values have been computed. Using the
table, probabilities for any other values of  and 2 can be
determined
Normal Distribution
The normal distribution for parameters values
 = 0, and 2 = 1 is called the standard normal
distribution. A r.v. that has a standard
distribution is called a standard normal random
variable (denoted by Z). The pdf of Z is:
f ( z) 
1
2
e
z2

2
,
  z  
Normal Distribution
The cumulative distribution of Z is
z
P( Z  z ) 
 f ( y)dy
and is denoted by  (Z)

Note: The N(0,1) Table returns the cumulative
probability up to z or (z)
Selecting a Distribution
 Theoretical prior knowledge
 Random arrival => exponential IAT
 Sum of large manufactures => Normal CLT
 Compare histogram with probability mass or
probability density