1. CS433: Modeling and Simulation
Al-Imam Mohammad bin Saud University
CS433: Modeling and Simulation
Lecture 04: Statistical Models
Dr. Anis Koubâa
09 October 2010

2. Goals for Today Review of the most common statistical models
Understand how to determine the empirical distribution from a statistical sample

3. Topics Discrete Probability Distributions Binomial Distribution
Bernoulli Distribution
Geometric Distribution
Discrete Poisson Distribution/Poisson Process
Continuous Probability Distribution
Uniform
Exponential
Normal
Empirical Distributions

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]
Probability Mass Function (PMF)
Probability Density Function (PDF)
Cumulative Distribution Function (CDF)

6. Expectation E(X): The expected value (the mean) of X Discrete
Continuous
Var(X) : The variance of X

7. Statistical Models : Properties
Context of applications
Probability distribution (PMF, PDF)
Cumulative Distribution Function (CDF)
Mean/Expected Value
Variance and Standard Deviation

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

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 < p = success probability
with

12. Binomial Distribution
Probability of k=2 successes in n=3 trials?
E1: 1 1 0
P(E1)= p(1 and 1 and 0) = p p (1-p) = p2 (1-p) or
E2: 1 0 1
P(E2)= p(1 and 0 and 1) = p (1-p) p = p2 (1-p) or
P(E3)= p(1 and 1 and 0) = p p (1-p) = p2 (1-p)
E3: 0 1 1
Probability of k=2 successes in n=3 trials?

13. End of Part 01
Administrative issues
Groups Formation

14. Modeling of Discrete Random Time
Geometric Distribution

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

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 17

18. Poisson Distribution
The Poisson distribution with the average rate parameter l
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.
PMF
CDF 18

19. Example: Poisson Distribution
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
The probability of having more than 3 cars entering the parking in one hour:
USE EXCEL/MATLAB
FOR COMPUTATIONS

20. Example: Poisson Distribution
Probability Mass Function
Poisson (l = 20 cars/hour)
Cumulative Distribution Function
Poisson (l = 20 cars/hour)

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.
Administrative issues
Groups Formation

22. Modeling of Random Number of Arrivals/Events
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”?
State Space = {SUNNY,
RAINNY}
Day
Day 1
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

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 l if:
describes the number of events in time interval
The mean and the variance are equal t
t+t
t+t+q

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

28. Inter-Arrival Times of a Poisson Process
CDF of Exponential distribution
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:
The Inter-arrival times of a Poisson process are exponentially distributed and independent with mean 1/l

29. Splitting and Pooling Splitting Pooling N1(t) ~ Poi[lp] lp l
N(t) ~ Poi(l1 + l2)
N1(t) ~ Poi[l1]
N2(t) ~ Poi[l2]
l1 + l2 l1 l2

30. Modeling of Random Number of Arrivals/Events
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 λ(t), 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 l1 l2 l3

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 l(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 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?

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 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)

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.
Administrative issues
Groups Formation

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:

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
PDF
CDF

39. Exponential Distribution
Modeling Random Time

40. Exponential Distribution
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:

41. Exponential Distribution
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).

43. Exponential Distribution
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).

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:

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 s2.

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 m and s, using the standard normal distribution:
Transformation of variables: let

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, F(1) is the complement of F (-1)

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)).

54. End of Chapter 4