ch 2. theory of probability
TRANSCRIPT
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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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