cs433: modeling and simulation

CS433: Modeling and Simulation

Dr. Anis Koubâa

Al-Imam Mohammad bin Saud UniversityAl-Imam Mohammad bin Saud University

09 October 2010

Lecture 04: Statistical Models

2

Goals for Today

2

Review of the most common statistical models Understand how to determine the empirical

distribution from a statistical sample

3

Topics

3

4

Discrete and Continuous Random Variables

5

Discrete versus Continuous Random Variables

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]

1

1. ( ) 0, for all

2. ( ) 1

i

ii

p x i

p x

i ip x P X x

Cumulative Distribution Function (CDF)

f xProbability Density Function

(PDF)

X

R

X

Rxxf

dxxf

Rxxf

X

in not is if ,0)( 3.

1)( 2.

in allfor , 0)( 1.

p X x

( )i

ix xp X x p x

0

xp X x f t dt

b

ap a X b f x dx

Probability Mass Function (PMF)

6

Expectation

i

ii xpxxE all

)()(

dxxxfxE )()(

2 2 2x xV X x f x dx x f x dx

2

2 2

1 1 1

n n n

i x i i i i ii i i

V X x p x x p x x p x

7

Statistical Models : Properties

8

Discrete Probability Distributions

Bernoulli Trials Binomial Distribution Geometric Distribution Poisson Distribution Poisson Process

9

Modeling of Random Events with Two-States

Bernoulli Trials Binomial Distribution

10

Bernoulli Trials Context: Random events with two possible values

Two events: Yes/No, True/False, Success/Failure Two possible values: 1 for success, 0 for failure. Example: Tossing a coin, Packet Transmission Status,

An experiment comprises n trial. Probability Mass Function (PMF): Probability in one trial

Bernoulli Process: n trials

, 1, 1, 2,...,

PMF: one trial ( ) 1 , 0 1 2

0, otherwise

j

j j

p x j n

p p x p q x , j , ,..., n

Expected Value: jE X p 2Variance : 1jV X p p

1 1 1 2, ,..., , ...n np X X X p X p X p X

11

Binomial Distribution Context: Number of successes in a series of n trials.

Example: the number of 'heads' occurring when a coin is tossed 50 times.

The number of successful transmissions of 100 packets Probability Mass Function (PMF)

Binomial distribution with parameters n and p, X ~ B(n,p)Probability of having k successes in n trial

where k = 0, 1, 2, ......., n k: number of success n = 1, 2, 3, ....... n: number of trials 0 < p < 1 p = success probability

1n kkn

P X k p pk

!

! !

n n

k k n k

with

Expected Value: E X n p 2Variance : 1V X n p p

12

Binomial Distribution

2

2 2

2 1 2 3 3 1

3! 61 1

2! 3 2 ! 2

p X p E p E p E p p

p p p p

!

! !

n n

k k n k

Probability of k=2 successes in n=3 trials?

E1: 1 1 0

E2: 1 0 1

E3: 0 1 1

P(E1)= p(1 and 1 and 0) = p p (1-p) = p2 (1-p)

P(E2)= p(1 and 0 and 1) = p (1-p) p = p2 (1-p)

P(E3)= p(1 and 1 and 0) = p p (1-p) = p2 (1-p)

Probability of k=2 successes in n=3 trials?

or

or

13

End of Part 01

Administrative issues• Groups Formation

14

Modeling of Discrete Random Time

Geometric Distribution

15

Context: the number of Bernoulli trials until achieving the FIRST SUCCESS. It is used to represent random time until a first transition occurs.

Geometric Distribution

1 , 0,1,2,...,PMF: ( )

0, otherwise

kq p k np X k

1Expected Value : E X

p

22 2

1Variance :

q pV X

p p

PMF

k

CDF: F 1 1k

X p X k p

16

Modeling of Random Number of Arrivals/Events

Poisson Distribution Poisson Process

17

Poisson Distribution Context: number of events occurring in a fixed period

of time Events occur with a known average rate and are independent

Possion distribution is characterized by the average rate The average number of arrival in the fixed time period.

Examples The number of cars passing a fixed point in a 5 minute

interval. Average rate: = 3 cars/5 minutes The number of calls received by a switchboard during

a given period of time. Average rate: =3 call/minutes The number of message coming to a router per second The number of travelers arriving to the airport for

flight registration

18

Poisson Distribution The Poisson distribution with the average rate

parameter

exp for 0,1, 2, ....PMF: !

0, otherwise

k

kp k P X k k

0

CDF: exp!

k i

i

F k p X ki

Expected value: E X

Variance: V X

PMF

CDF

the probability that there are exactly k arrivals in a certain period of time.

the probability that there are at least k arrivals in a certain period of time.

19

Example: Poisson Distribution

The number of cars that enter the parking follows a Poisson distribution with a mean rate equal to = 20 cars/hour

The probability of having exactly 15 cars entering the parking in one hour:

or

The probability of having more than 3 cars entering the parking in one hour:

1520

15 15 exp 20 0.05164915!

p P X

15 15 14 0.156513 0.104864 0.051649p F F

3 1 3 1 3

1 0 1 2 3

0.9999967

p X p X F

p p p p

USE EXCEL/MATLABFOR COMPUTATIONS

20

Example: Poisson Distribution

Probability Mass FunctionPoisson ( = 20 cars/hour)

Cumulative Distribution FunctionPoisson ( = 20 cars/hour)

0

20 exp 20

!

k i

i

F k p X ki

20exp 20

!

k

p X kk

21

Five Minutes Break

You are free to discuss with your classmates about the previous slides, or to refresh a bit, or to ask questions.


22


Poisson Distribution Poisson Process

23

Poisson Process

Process N(t) random variable N that depends on time t

Poisson Process: a Process that has a Poisson Distribution N(t) is called a counting poisson process

There are two types: Homogenous Poisson Process: constant average

rate Non-Homogenous Poissoon Process: variable

average rate

24

What is “Stochastic Process”?

Day 1

Day

Day 2 Day 3 Day 4 Day 5 Day 6 Day 7

THU FRI SAT SUN MON TUE WED

X(dayi): Status of the weather observed each DAY

State Space = {SUNNY, RAINNY}

" "1X Sday " "3X Rday

" "2X Sday

" "5X Rday

" "4X Sday

" "7X Sday

" "6X Sday

" " or " " : RANDOM VARIABLE that varies with the DAYiX S Rday

IS A STOCHASTIC PROCESSiX day

25

Examples of using Poisson Process

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 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.

26

(Homogenous) Poisson Process

Formally, a counting process is a (homogenous) Poisson process with constant average rate if:

describes the number of events in time interval

The mean and the variance are equal

for 0 and 0,1,2,...

( )PMF: ( ) exp

!

n

t n

p N t N t n p N nn

E N V N

, 0N

N t N t ,t t

t t+

( )N

t++

( )N

27

(Homogenous) Poisson Process Properties of Poisson process

Arrivals occur one at a time (not simultaneous)

has stationary increments, which means

The number of arrivals in time s to t is also

Poisson-distributed with mean has independent

increments

N t N s N t s

t s

, 0N

, 0N

28

Inter-Arrival Times of a Poisson Process 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 (A1, A2, …), where Ai is the elapsed time between arrival i and arrival i+1

The first arrival occurs after time t MEANS that there are no arrivals in the interval [0,t], As a consequence:

28

1 0 expp A t p N t t

1 11 1 expp A t p A t t

The Inter-arrival times of a Poisson process are exponentially distributed and independent with

mean 1/

Brian

Poi is not an abbreviation of Poisson that I have ever seen

29

Splitting and Pooling

Splitting

Pooling

N(t) ~ Poi()

N1(t) ~ Poi[p]

N2(t) ~ Poi[(1-p)]

p

(1-p)

N(t) ~ Poi(12)

N1(t) ~ Poi[]

N2(t) ~ Poi[]

1

2

30


Poisson Distribution Non Homogenous Poisson Process

31

Non Homogenous (Non-stationary) Poisson Process (NSPP)

The non homogeneous Poisson process is characterized by a VARIABLE rate parameter (t)λ , the arrival rate at time t. In general, the rate parameter may change over time.

The stationary increments, property is not satisfied

The expected number of events (e.g. arrival) between time s and time t is,

t

s t sλ(u) du

, : s t N t N s N t s

1 2 3

32

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 (t) defined as follows:

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?

80 cars/mn 8 960 cars/mn 9 1150 car/mn 11 1570 car/mn 15 17

if am t amif am t pm

tif am t pmif pm t pm

33

Example: Non-stationary Poisson Process (NSPP)

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.

We know that the number of cars between 12 pm and 16 pm, i.e. follows a Poisson distribution. During 12 pm and 16pm, the average number of cars is

Thus,

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)

11:30

8:00,11:30 8:00

9:00 11:00 11:30

8:00 9:00 11:00

80cars/mn 60mn 60cars/mn 120mn 50cars/mn 30mn 13500 cars

λ(u) du

λ(u) du λ(u) du λ(u) du

10000

1320016 12 10000 exp 13200

10000!p N N

16 12N N

12:00 16:00 180 50 60 70 13200 cars

34

Two Minutes Break

You are free to discuss with your classmates about the previous slides, or to refresh a bit, or to ask questions.


35

Continuous Probability Distributions

Uniform Distribution Exponential Distribution Normal (Gaussian) Distribution

36

Uniform Distribution

37

Continuous Uniform Distribution

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:

1,

PDF: ( )0, otherwise

a x bf x b a

0,

CDF: ( ) ,

1,

x a

x aF x a x b

b ax b

Expected value: 2

a bE X

2

Variance: 12

a bV X

38

Uniform Distribution

Properties is proportional to

the length of the interval

Special case: a standard uniform distribution U(0,1). Very useful for random

number generators in simulators

1 2p x X x

2 12 1

X XF X F X

b a

CDF

PDF

39

Exponential Distribution

Modeling Random Time

40


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 > 0 if its PDF and CDF are: exp , 0PDF: ( )

0, otherwise

x xf x

0

0, 0CDF: ( )

1 , 0x t x

xF x

e dt e x

1Expected value: E X

2

2

1Variance: V X

1exp , 0

( )

0, otherwise

xx

f x

0, 0

( )1 exp , 0

x

F x xx

41


1exp , 0

( ) 20 20

0, otherwise

xx

f x

0, 0

( )1 exp , 0

20

x

F x xx

µ=20 µ=20

42

Exponential Distribution 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:

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).

|p X s t X s p X t

43


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 s minutes is equal to the probability that you will wait t 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.”

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).

|p X s t X s p X t

44

Example: Exponential Distribution

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

The probability that the car repair time lasts between 2 to 3 hours is:

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:

33 1 3 1 3 1 1 exp 0.368

3p X p X F

3 3 2 0.145p X F F

12.5 1 | 2.5 1 1 1 exp 0.717

3p X X p X p X

45

Normal (Gaussian) Distribution

46

Normal Distribution

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, σ2) 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.

47

Normal or Gaussian Distribution

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:

where

The Normal distribution is denoted as This probability density function (PDF) is

a symmetrical, bell-shaped curve, centered at its expected value µ. The variance is 2.

2

1 1PDF: exp

22

xf x

2~ ,X N

1CDF: 1

2 2

xF x erf

2

0

2Error Function: exp

x

erf x t

48

Normal distribution

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).

49

Normal Distribution

Evaluating the distribution: Independent of and using the standard normal

distribution:

Transformation of variables: let

z t dtez 2/2

2

1)( where,

)()(

2

1

)(

/)(

/)( 2/2

xx

x z

dzz

dze

xZPxXPxF

~ 0,1Z N

XZ

50

Normal Distribution

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:

Using the symmetry property, (1) is the complement of (-1)

1587.0)1(2

1210)10(

F

51

Empirical Distribution

52

Empirical Distributions

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.

53

Empirical Distributions

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 Fn(x) based on sample X1, X2, …, Xn is a step function defined by

where I(A) is the indicator of event A. For a fixed value x, I(Xi≤x) is a Bernoulli random variable

with parameter p=F(x), hence nFn(x) is a binomial random variable with mean nF(x) and variance nF(x)(1-F(x)).

1

number of element in the sample 1 n

n ii

xF x I X x

n n

1 if

0 otherwisei

i

X xI X x

54

End of Chapter 4

cs433: modeling and simulation

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