CS433: Modeling and Simulation Dr. Anis Koubâa Al-Imam Mohammad bin Saud University 05 March 2010...

CS433: Modeling and Simulation

Dr. Anis Koubâa

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

05 March 2010

Lecture 03: Probability Review

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Goals of today

Review the fundamental concepts of probability

Understand the difference between discrete and continuous random variable

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Topics

Fundamental Laws of Probability Discrete Random Variable

Probability Mass Function (PMF) Continuous Random Variable

Probability Density Function (PDF) Cumulative Distribution Function (CDF) Mean and Variance

Probability Review

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Fundamentals

We measure the probability for Random Events How likely an event would occur

The set of all possible events is called Sample Space

In each experiment, an event may occur with a certain probability (Probability Measure)

Example: Tossing a dice with 6 faces The sample space is {1, 2, 3, 4, 5, 6} Getting the Event « 2 » in on experiment has a

probability 1/6

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The probability of every set of possible events is between 0 and 1, inclusive.

The probability of the whole set of outcomes is 1. Sum of all probability is equal to one Example for a dice: P(1)+P(2)+P(3)+

P(4)+P(5)+P(6)=1 If A and B are two events with no common outcomes,

then the probability of their union is the sum of their probabilities. Event E1={1}, Event E2 ={6} P(E1 U E2)=P(E1)+P(E2)

Probability

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Complementary Event of A is not(A)

P(A)=1-P(not A)

The probability that event A will not happen is

1-P(A).

Example

Event E1={1}

Probability to get a value different from {1} is 1-P(E1).

Complementary Event

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Event Union (U = OR) Event Intersection (∩= AND)

Joint Probability (A ∩ B)

The probability of two events in conjunction. It is the probability of both

events together.

Independent Events

Two events A and B are independent if

p A B p A p B p A B

p A B p A p B

Joint Probability

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p A B p A p B

Example on Independence

321

Case 1: Drawing with replacement of the ballThe second draw is independent of the first draw

1 1 11 2 1 2

3 3 9p E E p E p E

E1: Drawing Ball 1E2: Drawing Ball 2E3: Drawing Ball 3

Case 2: Drawing without replacement of the ballThe second draw is dependent on the first draw

1 1 11 2 1 2

3 2 6p E E p E p E

P(E1): 1/3P(E2):1/3P(E3): 1/3

Quiz: Show that in Case 2, we have 11 2

2p E E

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Conditional Probability Conditional Probability p(A|B) is the probability of

some event A, given the occurrence of some other event B.

If A and B are independent, then and If A and B are independent, the conditional probability of A, given

B is simply the individual probability of A alone; same for B given A.

p(A) is the prior probability; p(A|B) is called a posterior probability. Once you know B is true, the universe you care about shrinks to B.

|p A B

p A Bp B

|

p B Ap B A

P A

|p A B p A |p B A p B

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Example on Independence

321

Case 1: Drawing with replacement of the ballThe second draw is independent of the first draw

11| 2

3p E E

E1: Drawing Ball 1E2: Drawing Ball 2E3: Drawing Ball 3

Case 2: Drawing without replacement of the ballThe second draw is dependent on the first draw

11| 2

2p E E

P(E1): 1/3P(E2):1/3P(E3): 1/3

|p A B

p A Bp B

1 11 2 13 31| 2

12 33

p E Ep E E

p E

1 11 2 13 21| 2

12 23

p E Ep E E

p E

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Baye’s Rule

We know that

Using Conditional Probability definition, we have

The Bayes rule is:

|

|p B A p A

p A Bp B

p A B p B A

| |p A B P B p B A P A

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|p A E p A N

Law of Total Probability In probability theory, the law of total

probability means the prior probability of A, P(A), is equal to

the expected value of the posterior probability of A.

That is, for any random variable N,

where p(A|N) is the conditional probability of A given N.

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Law of Total Probability Law of alternatives: The term law of total

probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables.

if { Bn : n = 1, 2, 3, ... } is a finite partition of a

probability space and each set Bn is measurable,

then for any event A we have

or, alternatively, (using Rule of Conditional Probability)

nn

p A p A B

| nn

p A p A B p B

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Example: Law of Total Probability

1 2 3{3}

0 {3} {2,3,6} 0 {3}

p A p p A B p A B p A B

p p

{1,2,3,4,5,6,7}S

2 {2,3,6}B 3 {4,7}B 1 {1,5}B

Sample Space

Partitions

Event {3}A

Law of Total Probability

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 (mandatory to do it this week)

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A random variable

► Random variable is a measurable function from a sample space of events to the measurable space S of values of possible values of the variable

:

i i

X S

E x

Event Value

Each value/event has a probability of occurrence

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A random variable

► Random Variable is also know as stochastic variable.► A random variable is not a variable. It is a

function. It maps from the sample space to the real numbers.

► random variable is defined as a quantity whose values are random and to which a probability distribution is assigned.

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A random variable: Examples.

► The number of packets that arrives to the

destination

► The waiting time of a customer in a queue

► The number of cars that enters the parking

each hour

► The number of students that succeed in the

exam

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Random Variable Types► Discrete Random Variable:

► possible values are discrete (countable sample space, Integer values)

► Continuous Random Variable: ► possible values are continuous (uncountable

space, Real values)

: {1,2,3,4,...}

i i

X

E x

: 1.4,32.3

i i

X

E x

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Discrete Random Variable The probability distribution for discrete

random variable is called Probability Mass Function (PMF).

Properties of PMF

Cumulative Distribution Function (CDF)

Mean value

0 1 ip x 1 ii

p x

i ip x P X x

and

( )i

ix xp X x p x

1

n

X i ii

E X x p x

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Discrete Random Variable Mean (Expected) value

Variance

1

n

X i ii

E X x p x

2 22 2XV X E X E X E X E X

2

2

1 1

n n

i i i ii i

V X x p x x p x

General Equation

For Discrete RV

2

1

n

i x ii

V X x p x

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Discrete Random Variable: Example Random Variable: Grades of the students

21 1 0.2

10p P X

Student ID

1 2 3 4 5 6 7 8 9 10

Grade 3 2 3 1 2 3 1 3 2 2

42 2 0.4

10p P X

43 3 0.4

10p P X

Probability Mass Function

0 1 1p

0 2 1p

0 3 1p

PDF

Grade

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Discrete Random Variable: Example Random Variable: Grades of the students

Student ID

1 2 3 4 5 6 7 8 9 10

Grade 3 2 3 1 2 3 1 3 2 2

Probability Mass Function Property 1 2 3 1i

i

p x p p p

Cumulative Distribution Function

( )i

ix xp X x p x

2

2 ( ) 1 2 0.2 0.4 0.6i

ixp X p x p p

CDF

Grade

2

3 ( ) 1 2 3 1i

ixp X p x p p p

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Discrete Random Variable: Example Random Variable: Grades of the students

Student ID

1 2 3 4 5 6 7 8 9 10

Grade 3 2 3 1 2 3 1 3 2 2

The Mean Value

1 0 2 2 0 4 3 0 4 2 2i ii

x p x . . . .

2 210

i

Grade

.

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Continuous Random Variable The probability distribution for continuous

random variable is called Probability Density Function (PDF).

The probability of a given value is always 0 The sample space is infinite

For continuous random variable, we compute Properties of PDF

f x

X

R

X

Rxxf

dxxf

Rxxf

X

in not is if ,0)( 3.

1)( 2.

in allfor , 0)( 1.

0ip x x

p a x b

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Continuous Random Variable Cumulative Distribution Function (CDF)

Mean/Expected value

Variance

( )i

ix xp X x p x

0x

p X x f t dt

b

ap a X b f x dx

X E X x f x dx

2xV X x f x dx

2 2xV X x f x dx

and

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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)

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Continuous Random Variables: Example

Example: modeling the waiting time in a queue

People waiting for service in bank queue

Time is a continuous random variable Random Time is typically modeled as exponential

distribution is the mean value 1

exp , 0( )

0, otherwise

xx

f x

Exponential DistributionExp (µ)

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Continuous Random Variables: Example

We assume that with average waiting time of one customer is 2 minutes

otherwise ,0

0 x,2

1)(

2/xexf

PDF: f (time)

time

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Continuous Random Variables: Example

Probability that the customer waits exactly 3 minutes is:

Probability that the customer waits between 2 and 3 minutes is:

The Probability that the customer waits less than 2 minutes

3 /2

2

1(2 3) 0.145

2xP x e dx

3 /2

3

1( 3) (3 3) 0

2xP x P x e dx

1(0 2) (2) (0) (2) 1 0.632P X F F F e

145.0)1()1()2()3()32( 1)2/3( eeFFXP CDF

CDF

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Continuous Random Variables: Example

The mean of life of the previous device is:

To compute variance of X, we first compute E(X2):

Hence, the variance and standard deviation of the device’s life are:

/2 /2

0 00

1( ) 2

2

/2x xE X xe dx e dxxxe

82/22

1)(

0

2/

00

2/22

dxexdxexXE xx ex

2)(

428)( 2

XV

XV

Expected Value and Variance

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 (mandatory to do it this week)

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Variance

The standard deviation is defined as the square root of the variance, i.e.:

2X X V X s

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Coefficient of Variation

The Coefficient of Variance of the random variable X is defined as:

X

X

V XCV X

E X

End of lectureThere are some additional slides afterwards that

may help to explain some concepts

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Discrete Probability Distribution

The probability distribution of a discrete

random variable is a list of probabilities

associated with each of its possible values.

It is also sometimes called the probability

function or the probability mass function

(PMF) for discrete random variable.

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Probability Mass Function (PMF) Formally

the probability distribution or probability mass function (PMF) of a discrete random variable X is a function that gives the probability p(xi) that the random variable equals xi, for each value xi:

It satisfies the following conditions: 0 1 ip x

1 ii

p x

i ip x P X x

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Continuous Random Variable

A continuous random variable is one which takes an infinite number of possible values.

Continuous random variables are usually measurements. Examples include height, weight, the amount of sugar

in an orange, the time required to run a mile.

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Probability Density Function (PDF) For the case of continuous variables, we do

not want to ask what the probability of "1/6" is, because the answer is always 0...

Rather, we ask what is the probability that the value is in the interval (a,b).

So for continuous variables, we care about the derivative of the distribution function at a point (that's the derivative of an integral). This is called a probability density function (PDF).

The probability that a random variable has a value in a set A is the integral of the p.d.f. over that set A.

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Probability Density Function (PDF) The Probability Density Function (PDF) of a

continuous random variable is a function that can be integrated to obtain the probability that the random variable takes a value in a given interval.

More formally, the probability density function, f(x), of a continuous random variable X is the derivative of the cumulative distribution function F(x):

Since F(x)=P(X≤x), it follows that:

d

f x F xdx

b

a

F b F a P a X b f x dx

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Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF) is a function giving the probability that the random variable X is less than or equal to x, for every value x.

Formally the cumulative distribution function F(x) is defined

to be:

,

x

F x P X x

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Cumulative Distribution Function (CDF)

For a discrete random variable, the cumulative distribution function is found by summing up the probabilities as in the example below.

For a continuous random variable, the cumulative distribution function is the integral of its probability density function f(x).

,

( ) ( )

i i

i ix x x x

x

F x P X x P X x p x

b

a

F a F b P a X b f x dx

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Cumulative Distribution Function (CDF)

► Example ► Discrete case: Suppose a random variable X has

the following probability mass function p(xi):

► The cumulative distribution function F(x) is then:

xi 0 1 2 3 4 5

p(xi) 1/32 5/32 10/32 10/32 5/32 1/32

xi 0 1 2 3 4 5

F(xi) 1/32 6/32 16/32 26/32 31/32 32/32

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Mean or Expected Value

1

n

X i ii

E X x p x

X E X x f x dx

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Example: Mean and variance

When a die is thrown, each of the possible faces 1, 2, 3, 4, 5, 6 (the xi's) has a probability of 1/6 (the p(xi)'s) of showing. The expected value of the face showing is therefore:

µ = E(X) = (1 x 1/6) + (2 x 1/6) + (3 x 1/6) + (4 x 1/6) + (5 x 1/6) + (6 x 1/6) = 3.5

Notice that, in this case, E(X) is 3.5, which is not a possible value of X.

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Variance

The variance is a measure of the 'spread' of a distribution about its average value.

Variance is symbolized by V(X) or Var(X) or σ2.

The mean is a way to describe the location of a distribution,

the variance is a way to capture its scale or degree of being spread out. The unit of variance is the square of the unit of the original variable.

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Variance The Variance of the random variable X is

defined as:

where E(X) is the expected value of the random variable X.

The standard deviation is defined as the square root of the variance, i.e.:

2 22 2XV X E X E X E X E X

2X X V X s