Ch 2. Theory of Probability

8/12/2019 Ch 2. Theory of Probability

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Chapter 2

Theory of probability


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Sample space, sample points, events

Sample space is the set of all possible sample points

Example 0.Tossing a coin: = { H, T }

Example 1. Rolling die: = { 1, 2, 3, 4, 5, 6 }

Example 2.Number of customers in queue: = { 0, 1, 2, }

Example 3.Call holding time: = { }

EventsA, B, C, are (measurable) subsets of the sample space

Example 1. Even numbers of a die: A = { 2, 4, 6 }

Example 2. No customers in queue: A = { 0 }

Example 3. Call holding time greater than 3.0 (min) : A = { }

Denote by the set of all events A

Sure event : The sample space itself

Impossible event:The empty set


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Combination of events

Un ion A or B :

Intersection A or B :

Complementnot A :

Events A and B are disjointif

A set of events { B1, B2, - is a partitionof event A if

i. for allIi.

BA

}BrA|{BA o

}BndA|{BA a

}A|{A c

BA

ji

ABii B1

B2

B3


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Probability

Probability of event is denoted by P(A), P(A) * 0, 1 +

Probabilitymeasure P is thus

A real-valued set function define on the set of events ,

P : [0,1]

Properties

i. 0 P(A) 1

ii. P() = 0iii. P() = 1

iv. P(A) = 1P(A)


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v.

vi.vii.

viii.

B)P(A-P(B)+P(A)=B)P(A

P(B)+P(A)=B)P(AdisjointareBandA

)(P(A)Aofpartitionis}{ i ii BPB

)()( BPAPBA

A

B


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Conditional probability

Assume that P(B) > 0

Definition :The conditional probability of event A given

that event B occurred is defined as

It follow that

)(

)|()|(

BP

BAPBAP

)|()()|()()|( BAPAPBAPBPBAP


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Bayes theorem

Let {Bi- be partition of the sample space

Assume that P(A) > 0 and P(Bi) > 0 for all I, Then by (slide 6)

Furthermore, by the theorem of total probability (slide 7),we get

This is Bayesstheorem Probabilities P(Bi) are called a prioriprobabilities of event

Bi

Probabilities P( Bi|A ) are called a posteriori probabilities

of events Bi (given that the event A occured )

)(

)|()(

)(

)()|(

AP

ABPBP

AP

BAPABP iii

)|()(

)|()()|(

jjj

iii

BAPBP

BAPBPABP


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Random Variables

Definition :real-valued random variable X is a real-valued and measurable function defined on thesample space

Each sample point Is associated with a realnumber

Measurabilitymeans that all sets of type

Belong to the set of events, that is

The probability of such an event is denoted by

:,X

})(|{:}{ xXxX

}{ xX

}{ xXP


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Example

A coin is tossed three times

Sample space :

Let X be the random variable that tells the

total number of tails in these three

experiments :

HHH HHT HTH THH HTT THT TTH TTT

0 1 1 1 2 2 2 3

1,2,3}=iT},{H,|),,{(= i321

)(x


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Indicators of events

Lests being an arbitrary event

Definitioan : Theindicator of event A is a random

variable defined as follows:

Clearly:

A

A

AA

,0

,1

)(}11{ APP A )(1)(}01{

0 APAPP A


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Cumulative distribution

Definition :the cumulative distribution function (cdf) of a

random variable X is a function define as follow:

Cdf determain the distributionof the random variable, That is : the probabilities P X B -, where

Properties

i. Fx is non-decreasing

ii. Fx is continuous form the right

iii. Fx (-) = 0

iv. Fx () = 1

]1,0[: xF

)(:)( xXPxFx

}{ BXandB

0

Fx(x)

1

x


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Statistical independence of

randomvariables

Definition :Random variables X and Y independent if for

all x and y

Definition : Random variables X1, , Xn are (totally)

independentif for all i and xi

y}}P{YxXP{=y}Yx,P{X

}xP{X}...xXP{=}xX...,,xP{X nn11nn11


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Maximum and minimum of

independent random variable

Let the random variables X1, , Xnbe independent

Denote : X max: = max { X1, , Xn}. Then

Denote : X min: = min { X1, , Xn}. Then

x}Xn,,xXP{=}xP{X 1max

x}P{Xn,

},xXP{= 1

x}Xn,,xXP{=}xP{X1

min

x}P{Xn,},xXP{= 1


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Discrete random variables

Definition :set is called discreteif it is

Finite, A= { X1, , Xn} , or

Countably infinite, A= { X1, X2 , , -

Definition :random variable X is dicreteif there is a discretesetsuch that

It follow that

The set Sx is called the value set

A

xS

1}{ xSXP

xSxallfor00}{ xXP

xSxallfor00}{ xXP


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Point probabilities

Let X be a discrete random variable

The distribution of X is determined by the point

probabilitiespi ,

Definition :the probability mass function(pmf) of X is a

function

Cdf is in this case a step function:

X

Xii

x S x0,

Sx= x,p

}x=P{X:(x)p

[0,1]=px

Xiii Sx},x=P{X:p

xxi

i

ii

p}xP{X:(x)Fx


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Probabilitas mass function (pdf) cumulative distribution function (cdf)

Example

X1

X2

X3

X4

X1

X2

X3

X4

Px(X) Fx(X)

x

1

x

1

}x,x,x,{x=S 4321X


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Impendence of discrete random

variables

Discrete random variables X and Y are independent if and

only if for all

}{}{],[iiii

yYPxXPyYxXP


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Expectation

Definition :the expectation(mean value ) of X is defined by

Note 1 : the expectaftion exists only if

Note 2 : if , then we may denote E*X+ =

Properties :

i.ii.

iii.

i

Sx

i

Sx

x

SxSx

x xpxxpxxXPxxXPXEXXXX

..)(.][.][][

|| iii xp

|| iii xp

cE[X]=E[cX]Rc E[Y]E[X]=Y]E[X

E[X]E[Y]=[XY]EtindependenYandX


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Variance

Definition : the varianceof X is defined by

Useful formula (prove!) :

Properties :

i.

ii.

]E[X])-E[(X:Var[X]:[X]D:22

X2

222E[X]-]E[X[X]D

[X]Dc=[cX]Dc222

[Y]D+[X]D=Y]+[XDtindependenYandX 222


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Covariance

Definition :the covariancebetween X and Y is defined by

Useful formula (prove!) :

Properties :

i. Cov[X,X] = Var[X]

ii. Cov[X,Y] = Cov[Y,X]iii. Cov[X+Y,Z] = Cov[X,Z] + Cov[Y,Z]

iv. X and Y independent

E[Y])]-(Y)E[X]-E[(X:Y]Cov[X,:=XY2

E[X]E[Y]-E[XY]Y]Cov[X,

0Y]Cov[X,


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Other distribution related parameters

Definition :the standard deviationof X is defined by

Definition :the coefficient of varition of X is defined by

Definition :thek

th momentof X is defined by

Var[X]=[X]D:D[X]:2

x

][

][:C[X]:

xYE

XDc

]E[X:k

x)( k


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THANK YOU.

Ch 2. Theory of Probability

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