Al-Imam Mohammad Ibn Saud University
1
27 Oct 2008
CS433
Modeling and Simulation
Lecture 04
http://10.2.230.10:4040/akoubaa/cs433/




2
2

 Discrete Random Variable
 Continuous Random Variable
 Discrete Probability Distributions
 Binomial Distribution
 Bernoulli Distribution
 Discrete Poisson Distribution
 Continuous Probability Distribution
 Uniform
 Exponential
 Normal
 Weibull
 Lognormal
 Empirical Distributions 3
3

Discrete and Continuous Random Variables
4






X is a discrete random variable if the number of possible values of X (the sample space) is finite .
Example: Consider jobs arriving at a job shop.
 Let X be the number of jobs arriving each week at a job shop.


R x
= possible values of X (range space of X)
p(x i
) = probability the random variable is x i
= {0,1,2,…}
= p(X = x i
)
The collection of pairs [x i distribution of X ,
, p(x i
)], i = 1,2,…, is called the probability
p(x i
) is called the probability mass function (PMF) of X .
Characteristics of the PMF: p(x i
), i = 1,2, … must satisfy: p x i
)

0, for all i
2.
 i


1 p x i

5



X is a continuous random variable if its range space R x collection of intervals.
is an interval or a


The probability that X lies in the interval [a,b] is given by:
P ( a

X
 b )
  b f ( x ) dx a
Where f(x) is the probability density function (PDF) .
Characteristics of the PDF : f(x) must satisfies:
1.
f ( x )

0 , for all x in R
X
2.
R

X
3.
f f ( x ) dx
( x )


1
0 , if x is not in R
X
Properties
1.
P ( X
2 .
P ( a


X x
0
)

 b )
0 , because
 x x
0
0

P ( a  X f
 b )
( x ) dx

P ( a


0
X  b )

P ( a  X  b )
6

Discrete Random Variable
Continuous Random Variable
Finite Sample Space e.g. {0, 1, 2, 3}
Infinite Sample Space e.g. [0,1], [2.1, 5.3]
Probability Mass Function (PMF)
     x i p x i
)

0, for all i
2.
 i


1 p x i


Probability Density Function (PDF)
 
1.
f ( x )

0 , for all x in R
X
2.
R
X
 f ( x ) dx

1
3.
f ( x )

0 , if x is not in R
X
Cumulative Distribution Function (CDF)
  x

  x
   x i
 x p x i
)

  x
 

X
 b

 x

  a
  
0 b f
 
7

You are free to discuss with your classmates about the previous slides, or to refresh a bit, or to ask questions.
Administrative issues
•
Groups Formation
•
Choose a “class coordinator”
8


The expected value (the mean) of X is denoted by E(X)

If X is discrete E ( x )
  all i x i p ( x i
)

If X is continuous
E ( x )
  
  xf ( x ) dx



A measure of the central tendency
The variance of X is denoted by V(X) or var(X) or s
2



Definition:
Also,
 
 


E X

2




 
2
  
2
A measure of the spread or variation of the possible values of X around the mean
The standard deviation of X is denoted by s

Definition: square root of V(X)

Expressed in the same units as the mean
9

Example: Continuous Random Variables



Example: modeling the lifetime of a device
Time is a continuous random variable
Random Time is typically modeled as exponential distribution
We assume that with average lifetime of a device is 2 years f ( x )



1
2
0 e
 x / 2
, x

0
, otherwise

Probability that the device’s life is between 2 and 3 years is:
P ( 2
 x

3 )

1
2

2
3 e
 x / 2 dx

0 .
14
10

Example: Continuous Random Variables

Cumulative Distribution Function : A device has the CDF:
F ( x )

1
2

0 x e
 t / 2 dt

1
 e
 x / 2

The probability that the device lasts for less than 2 years:
P ( 0

X

2 )

F ( 2 )

F ( 0 )

F ( 2 )

1
 e

1 
0 .
632

The probability that it lasts between 2 and 3 years:
P ( 2

X

3 )

F ( 3 )

F ( 2 )

( 1
 e

( 3 / 2 )
)

( 1
 e

1
)

0 .
145
11

Example: Continuous Random Variables
Expected Value and Variance


Example: The mean of life of the previous device is:
( )

1
2

0

 x / 2 xe dx

 xe
 x /2
0

 
0

 x / 2 e dx

2
To compute variance of X, we first compute E(X 2 ):
E ( X
2
)

1
2

0
 x
2 e
 x / 2 dx

 x 2 e
 x / 2
0

 
0
  e x / 2 dx

8

Hence, the variance and standard deviation of the device’s life are:
V ( X )

8

2
2 
4 s 
V ( X )

2
12

Discrete Probability Distributions





13


Discrete random variables are used to describe random phenomena in which only integer values can occur.

In this section, we will learn about:

Bernoulli trials and Bernoulli distribution

Binomial distribution

Geometric and negative binomial distribution

Poisson distribution
14
14

Modeling of Random Events with Two-States


15


In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes , " success " and
" failure ".

In practice it refers to a single experiment, which can have one of two possible outcomes. These events can be phrased into “yes” or “no” questions:

Did the coin land heads?

Was the newborn child or a girl?

Were a person's eyes green?
16



Consider an experiment consisting of n trials, each can be a
success or a failure.


Let X j and X j
= 1 if the j th experiment is a success
= 0 if the j th experiment is a failure
The Bernoulli distribution (one trial):
PMF: p x j
)
 p x j
)



1 p , p q , x x j j


1,
0 j
0, otherwise


1, 2,...,
1 2 n
Expected Value:
 j
  p Variance : V  X j
 s 2 p

1 p


Bernoulli process

It is the n Bernoulli trials where trials are independent:

1
, X
1
,..., X n
,
      
...
 
17



A binomial random variable is the number of successes in a series of n trials .

Example: the number of 'heads' occurring when a coin is tossed 50 times.
A discrete random variable X is said to follow a Binomial distribution with parameters n and p , written X ~ Bi(n,p) or
X ~ B(n,p) if it has the probability distribution:

 where
 k
 


x = 0, 1, 2, ......., n
n = 1, 2, 3, ....... p k
 p


p = success probability; 0 < p < 1
Where
 
 
 k !
 n n
!
 k

!
Expected Value:
  n p
Variance :
 
 s 2 n p

1 p

18


The trials must meet the following requirements:
 the total number of trials is fixed in advance;
 there are just two outcomes of each trial; success and failure ;
 the outcomes of all the trials are statistically independent ;
 all the trials have the same probability of success.
19
19


The number of successes in n Bernoulli trials, X , has a binomial distribution.
p ( X
 k )




 n k

 p k q n
 k
, k

0 , 1 ,
0 , otherwise
2 ,..., n
The number of outcomes having the required number of successes and failures
Probability that there are x successes and
(n-x) failures

The formula can be understood as follows: we want k successes
(p k ) and n − k failures (1 − p) n − k . However, the k successes can occur anywhere among the n trials, and there are C(n, k) different ways of distributing k successes in a sequence of n trials.
20

Administrative issues
•
Groups Formation
•
Choose a “class coordinator”
21


22

Geometric Distribution

Geometric Distribution represents the number X of Bernoulli trials to achieve the FIRST SUCCESS .

It is used to represent random time until a first transition occurs
PMF
PMF: (
 k )
 q k

1 p , k

0,1, 2,...,
 0, otherwise n
CDF: F
     k

1

1 p
 k
Expected Value :
 

1 p
Variance :
 
 s 2  q p
2

1
 p p
2 k
23


24

Negative Binomial Distribution


The negative binomial distribution is a discrete probability distribution that can be used to describe the distribution arising from an experiment consisting of a sequence of independent trials, subject to several constraints.
Firstly each trial results in success or failure, the probability of success for each trial, p , is constant across the experiment and finally the experiment continues until a fixed number of successes have been achieved.

Negative Binomial Distribution


The number of Bernoulli trials, X , until the k th success
If X is a negative binomial distribution with parameters p and r , then:

,
   k


 k r 1

 p r

1 p
 k
, k

1, 2,3...
 k

0, otherwise
Expected Value :
 
 r
1
 p p
Variance :
 
 s 2  r
 p
 p
2
25

Modeling of Random Number of
Arrivals/Events


26



 the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event.
Poisson random variable represents the count of the number of events that occur in a certain time interval or spatial area.
Example:

The number of cars passing a fixed point in a 5 minute interval,

The number of calls received by a switchboard during a given period of time.

The number of message coming to a router in a given period of time
27


A discrete random variable X is said to follow a Poisson distribution with parameter l
, written X ~ Po( l ), if it has probability distribution
PMF:
  k
  l k k !
exp

The PMF represents the probability that there are k arrivals in a certain period of time.
 where


X = 0, 1, 2, ..., n l > 0 is called the arrival rate.
28



Poisson distribution describes many random processes quite well and is mathematically quite simple.
The Poisson distribution with the parameter l is characterized by:
PMF:
     k


 l k k
!
exp
 
for

0, otherwise
0,1, 2, ....
PMF
CDF:
     k
  i k 

0 l i i !
 exp
Expected value:
 
 l
Variance:
 
 l
CDF
29


The following requirements must be met in the Poisson Distribution:
 the length of the observation period is fixed in advance;
 the events occur at a constant average rate;
 the number of events occurring in disjoint intervals are statistically independent.
30


The number of cars that enter the parking follows a Poisson distribution with a mean rate equal to l = 20 cars/hour

The probability of having exactly 15 cars entering the parking in one hour: or p
 



15


20
15
15!
 exp


20


0.051649
p
 

F
 

F
 
  
0.051649

The probability of having more than 3 cars entering the parking in one hour:
 
3

1 p
 
3

1 F
   p
   p
   p
 

0.9999967
USE EXCEL/MATLAB

Probability Mass Function
Poisson ( l
= 20 cars/hour)
  k
 
20 k k !
 exp
  
Cumulative Distribution Function
Poisson ( l
= 20 cars/hour)
     k
  i k 

0
20 i i !
 exp
 
20

32

You are free to discuss with your classmates about the previous slides, or to refresh a bit, or to ask questions.
Administrative issues
•
Groups Formation
33

Modeling of Random Number of
Arrivals/Events


34





Wikipedia: A Poisson process , named after the French mathematician Siméon-Denis Poisson (1781 – 1840), is the
stochastic process in which events (e.g. arrivals) occur continuously and independently of one another .
Formal Definition: The Poisson Process is a counting function { N( t ), t ≥0} where N( t ) is the number of events that have occurred up to time t , i.e. in the interval [0, t ].
Fact: The number of events between time a and time b is given as
N(b) − N(a) and has a Poisson distribution .
The Poisson process is a continuous-time process : Time is continuous

Its discrete-time counterpart is the Bernoulli process
 Bernoulli process is a discrete-time stochastic process consisting of a sequence of independent random variables taking values over two symbols. 35






The number of web page requests arriving at a server may be characterized by a Poisson process except for unusual circumstances such as coordinated denial of service attacks.
The number of telephone calls arriving at a switchboard, or at an automatic phone-switching system, may be characterized by a
Poisson process.
The number of raindrops falling over a wide spatial area may be characterized by a spatial Poisson process.
The arrival of "customers" is commonly modelled as a Poisson process in the study of simple queueing systems.
The execution of trades on a stock exchange, as viewed on a tick by tick basis, is a Poisson process.
36



The homogeneous Poisson process is characterized by a CONSTANT rate parameter λ , also known as intensity , such that the number of events in time interval

, l t
 t  follows a Poisson distribution with

N
 
Poisson process with mean rate l if:

0
 for t

0 and n

0,1, 2,...
PMF:
     n
   n
 
( l t
) n n !
exp
 


    describes the number of events in time interval

,
The mean and the variance are equal

 
V

N
  l t
 t 
37


Properties of Poisson process



Arrivals occur one at a time (not simultaneous)
N
  
0
 has stationary increments , which means
        s

The number of arrivals in time s to t is also Poisson-

 distributed with mean
N
 

0
 l  s
 has independent increments
38

CDF of Exponential distribution
39


Inter-arrival time: time between two consecutive arrivals

The inter-arrival times of a Poisson process are random.
What is its distribution ?
Consider the inter-arrival times of a Poisson process (A
1 elapsed time between arrival i and arrival i+1
, A
2
, …) , where A i is the

The first arrival occurs after time t MEANS that there are no arrivals in the interval [0,t] , As a consequence:

1
 t
 
   
0

 exp
 t


1
 
1 t
  t

The Inter-arrival times of a Poisson process are exponentially distributed and independent with mean 1/ l 39



Splitting


A Poisson process can be split into two Poisson processes : The first with a probability p and the second with probability 1-p .
 

N
1
 

N
2

  rates l  p and l   p where

N(t) ~ Poi( l )
N
1
  l and
N
2 t l p
  are both Poisson processes with
N1(t) ~ Poi[ l p] l (1-p)
N2(t) ~ Poi[ l (1-p)]
Pooling


The summation of two Poisson processes is a Poisson process
N
1
  
N
2
      
N1(t) ~ Poi[ l
1
] l
1 l
1
 l
2
N(t) ~ Poi( l
1
 l
2
)
N2(t) ~ Poi[ l
2
] l
2
2
40

Modeling of Random Number of
Arrivals/Events


41

Non Homogenous (Non-stationary) Poisson Process (NSPP)

The non homogeneous Poisson process is characterized by a VARIABLE rate parameter λ(t) , the arrival rate at time t. In general, the rate parameter may change over time. l
1 l
2 l
3

The stationary increments, property is not satisfied

, :
 

 


 s


The expected number of events (e.g. arrival) between time s and time t is l   s t
42

Example: Non-stationary Poisson Process (NSPP)

The number of cars that cross the intersection of King Fahd Road and Al-Ourouba
Road is distributed according to a non homogenous Poisson process with a mean l (t) defined as follows: l 





80 cars/mn if 8 am
 
 
9 am
60 cars/mn if 9 am t 11 pm
50 car/mn
70 car/mn if if
11
15 am pm
 
 
15
17 pm pm




Let us consider the time 8 am as t=0.
Q1. Compute the average arrival number of cars at 11H30?
Q2. Determine the equation that gives the probability of having only 10000 car arrivals between 12 pm and 16 pm.
Q3. What is the distribution and the average (in seconds) of the inter-arrival time of two cars between 8 am and 9 am?
43

Example: Non-stationary Poisson Process (NSPP)

Q1. Compute the average arrival number of cars at 11H30?
l
8:00,11:30
 11:30 
8:00




9:00  11:00  11:30
λ(u) du  λ(u) du 
8:00 9:00 11:00
80cars/mn 60mn
       
13500 cars


Q2. Determine the equation that gives the probability of having only 10000 car arrivals between 12 pm and 16 pm.
    follows a Poisson distribution. During 12 pm and 16pm, the average number of cars is l
12:00

16:00

180 50 60 70 13200 cars
Thus,
  
N
  
10000


13200

10000
10000!
 exp
 
13200

Q3. What is the distribution and the average (in seconds) of the inter-arrival time of two cars between 8 am and 9 am? (Homework) 44

You are free to discuss with your classmates about the previous slides, or to refresh a bit, or to ask questions.
Administrative issues
•
Groups Formation
45

Continuous Probability Distributions





46


Continuous random variables can be used to describe random phenomena in which the variable can take on any value in some interval.

In this section, the distributions studied are:

Uniform

Exponential

Normal

Weibull

Lognormal
47

Uniform Distribution
48



The continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable
A random variable X is uniformly distributed on the interval [a,b] ,
U(a,b) , if its PDF and CDF are:


 b
1
 a
, a
  b

0, otherwise
Expected value:
 

2

0, x a

 x a
, a
 x b

 b a
1, x
 b
Variance:
 

  2
12
49


Properties


1

X
 x 2
 is proportional to the length of the interval
     
X
2

X b
 a
1

Special case: a standard uniform distribution U(0,1).

Very useful for random number generators in simulators
PDF
CDF
50

Exponential Distribution

51



The exponential distribution describes the times between events in a Poisson process, in which events occur continuously and independently at a constant average rate.
A random variable X is exponentially distributed with parameter m1/l > 0 if its PDF and CDF are:
 l exp
 l x

, x

0
0, otherwise
 

1 m
 exp


 x m


, x

0
0, otherwise
 


0, x

0
0 x l e
 l t dt e
 l x
, x

0



0, x

0


 x m


, x

0
Expected value:
 

1 l
 m
Variance:
 
 l
1
2
 m 2
52

µ=20 µ=20
( )
 

1
20
 exp
 x
20
, x

0
0, otherwise
0, x

0

  x
20
, x

0
53




The memoryless property : In probability theory, memoryless is a property of certain probability distributions: the exponential distributions and the geometric distributions , wherein any derived probability from a set of random samples is distinct and has no information (i.e. "memory") of earlier samples.
Formally, the memoryless property is:
For all s and t greater or equal to 0:
  
|
 s
    t

This means that the future event do not depend on the past event , but only on the present event

The fact that Pr(X > 40 | X > 30) = Pr(X > 10) does not mean that the events X > 40 and X > 30 are independent; i.e. it does not mean that
Pr(X > 40 | X > 30) = Pr(X > 40) .
54



The memoryless property : can be read as “ the probability that you will wait more than s+t minutes given that you have already been waiting t minutes is equal to the probability that you will wait s minutes .”
In other words “ The probability that you will wait s more minutes given that you have already been waiting t minutes is the same as the probability that you had wait for more than s minutes from the beginning .”
  
|
 s
    t


The fact that Pr(X > 40 | X > 30) = Pr(X > 10) does not mean that the events
X > 40 and X > 30 are independent; i.e. it does not mean that
Pr(X > 40 | X > 30) = Pr(X > 40) .
55


The time needed to repair the engine of a car is exponentially
distributed with a mean time equal to 3 hours.

The probability that the car spends more than 3 hours in reparation


3

1


3

1 F
 
1


1 exp



3
3




0.368

The probability that the car repair time lasts between 2 to 3 hours is: p
 
X

3
 
F
  
F
  
0.145

The probability that the repair time lasts for another hour given it has been operating for 2.5 hours:
Using the memoryless property of the exponential distribution, we have:
  
X

2.5

 
1

1

1
 exp 

1
3


0.717
56

Normal (Gaussian) Distribution
57





The Normal distribution , also called the Gaussian distribution , is an important family of continuous probability distributions, applicable in many fields.
Each member of the family may be defined by two parameters, location and scale : the mean ("average", μ ) and variance
(standard deviation squared, σ 2) respectively.
The importance of the normal distribution as a model of quantitative phenomena in the natural and behavioral sciences is due in part to the Central Limit Theorem.
It is usually used to model system error (e.g. channel error), the distribution of natural phenomena, height, weight, etc.
58




A continuous random variable X , taking all real values in the range (-∞,+∞) is said to follow a Normal distribution with parameters µ and σ if it has the following PDF and CDF:
PDF: f
  
1 s 
2

 exp

 
1
2

 x
 s m 

2


CDF:
 
2

 1 erf

 x s 
 m
2



 where
Error Function: erf
  
2


 x
0 exp
The Normal distribution is denoted as X ~ N

2

This probability density function (PDF) is
 a symmetrical, bell-shaped curve,

 centered at its expected value µ.
The variance is s 2 .
59



Example
The simplest case of the normal distribution, known as the Standard Normal
Distribution , has expected value zero and variance one. This is written as
N(0,1) .
60


Evaluating the distribution:

Independent of m and s, using the standard normal distribution :
Z ~ N
 

Transformation of variables: let
Z

X s
 m
F ( x )

P

X
 x


P

Z
 x
 s m


( 
  x
 m
) / s 1
2

( 
  x
 m
) / s

( z ) dz e
 z
2
/ 2 dz
 
( x
 s m
)
, where

( z )
 
 z

1
2
 e
 t
2
/ 2 dt
61


Example: The time required to load an oceangoing vessel, X, is distributed as N(12,4)

The probability that the vessel is loaded in less than 10 hours:
F ( 10 )
 

10

2
12
 
(

1 )

0 .
1587

Using the symmetry property,  (1) is the complement of  (-1)
62

Other Distributions


63


A random variable X has a Weibull distribution if its pdf has the form: f ( x )



 b a x a
 b 
1 exp
0 ,



 x a
 b 


, x
  otherwise


3 parameters:


Location parameter: u,
Scale parameter: b , b  0
(
 

 
)

Shape parameter. a,  0
Example: u = 0 and a = 1:

Lifetime of objects
When b
= 1 ,
X ~ exp( l
= 1/ a
)
64


A random variable X has a lognormal distribution if its pdf has the form: f ( x )




2
1
π σx
0, exp




 ln
2 x
σ

2
μ

2




, x  0 otherwise


Mean E(X) = e m + s 2
/2
Variance V(X) = e 2m + s 2
/2 ( e s 2
- 1)
 m
=1, s
2=0.5,1,2.

Relationship with normal distribution


When Y ~ N( m , s 2 ), then X = e Y ~ lognormal( m , s 2 )
Parameters m and s 2 are not the mean and variance of the lognormal
 general reliability analysis
 65
65

Empirical Distribution
66


An Empirical Distribution is a distribution whose parameters are the observed values in a sample of data.

May be used when it is impossible or unnecessary to establish that a random variable has any particular parametric distribution.

Advantage : no assumption beyond the observed values in the sample.

Disadvantage : sample might not cover the entire range of possible values.
67



In statistics, an empirical distribution function is a cumulative probability distribution function that concentrates probability 1/n at each of the n numbers in a sample.
Let
X1, X2, …, Xn be iid random variables in with the CDF equal to F(x).

The empirical distribution function F n is a step function defined by
(x) based on sample
X1, X2, …, Xn
F n
   number of element in the sample n
 x

1 n i n 

1
 i
 x
 where I(A) is the
 indicator of event
For a fixed value x, I(X i n
A.
 i
 x



1 if

X i

0 otherwise x
(x) is a binomial random variable with mean nF(x) and variance nF(x)(1-F(x)).

≤x) is a Bernoulli random variable with parameter p=F(x) , hence nF
68

69