Download Chapter 1 Axioms of Probability

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Chapter 1
Axioms of Probability
§ 1.1 Introduction
• Relative frequency interpretation of probability
Example
Ball drawing with replacement
There are s balls in a box, labeled 1,2,...,s. Let N(k) be the
number of ball k being drawn in 20 trials and s=3.
E.g.
N(1)/20 = 0.25,
N(2)/20 = 0.4,
N(3)/20 = 0.35.
As the number of trials approaches infinity, we say
p1 = lim N(1)/n,
n-> 
p2 = lim N(2)/n,
n-> 
p3 = lim N(3)/n .
n-> 
Example
Decay of isotope
Let N(t) be the number of isotope atoms left at time t,
then
d N(t)/dt =
- N(t) and N(0) = N.
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Shi-Chung Chang & Tzi-Dar Chiueh, Spring 2006
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N(t)/N
= e -t
t0
,
The fraction of atoms that decay in interval [t0, t1] is
e -t0 -
e -t1
,
which is the probability of an atom decays in interval
[t0, t1], 0  t0  t1 .
• Model
* A model is a simplified, approximate representation of
a physical system. Mathematical models are used when
the observed phenomena have measurable properties.
* We use a deterministic model to describe experiments
whose outcomes are exact. A probability model can be
used to represent random experiments whose outcomes
varies in an unpredictable way even if those experiments
are repeated under the same conditions.
• Modeling procedure
* The first thing that must be done when modeling a
random experiment is to determine all possible
"outcomes" and choose a set S (sample space) of which
each elements x corresponds to one and only one
outcome.
* Next we must find "events" that correspond to subsets
of S and assign probability measures to all events.
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Shi-Chung Chang & Tzi-Dar Chiueh, Spring 2006
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§ 1.2
Sample Space and Events
• Definitions
* Sample Space (S): the set of all possible outcomes in an
experiment.
* A sample space is a set and thus can be continuous,
discrete or neither. A discrete sample space can have
either finite or infinite number of elements (outcomes).
* Sample Point: an outcome.
* Events: subsets of the sample space.
• Set (Event) Properties:
(1) E F: x  E, x  F.
(2) E = F: E  F and F  E.
(3) E  F or EF: intersection of E and F.
(4) E  F: Union of E and F.
(5) Ec = S - E = { x|x  S and x  E}.
(6) Certain event: S.
(7) Impossible event: 
(8) Mutually exclusive events: E  F = 
(9) Mutually exclusive event set: ij, Ei Ej = 
• Closedness with respect to Union and intersection
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Shi-Chung Chang & Tzi-Dar Chiueh, Spring 2006
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• Set Properties:
(Ec)c = E, E  Ec = S, EEc = 
(1) For any event E,
(2) Commutative law: E  F = F  E, EF = FE.
(3) Associative law: (E  F)  G = E  (F  G) and
(EF)G = E(FG)
(4) Distributive law: (EF)  H = (E  H)(F  H) and
(E  F)H = EH  FH
(5) De Morgan's first law
( Ei ) c
 Eic
=
i
i
(8) De Morgan's second law
(  Ei ) c
i
=
 Eic
i
* See Example 1.8 for proof.
• Examples of probability spaces
* Ball drawing from a box with replacement
S
= { 1, 2, ....., s } ;
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Shi-Chung Chang & Tzi-Dar Chiueh, Spring 2006
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* Decay of isotope
S
= [ 0, ) ;
* Other examples
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Shi-Chung Chang & Tzi-Dar Chiueh, Spring 2006
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Shi-Chung Chang & Tzi-Dar Chiueh, Spring 2006
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§ 1.3 Axioms of Probability
• Three axioms in probability theory:
* Axiom 1: P(A)  0.
* Axiom 2: P(S) = 1.
* Axiom 3: If A1, A2, ... is a sequence of mutually
exclusive events, then
 Ai ) = 
 P( Ai)
P(
i=1
i=1
• Theorem 1.1: P() = 0.
• Theorem 1.2: If A1, A2, ..., An is a sequence of mutually
exclusive events, then
n
n
P ( Ai ) =  P( Ai)
i=1
i=1
• From Theorem 1.2, we have
P(A  Ac) = P(S) = 1 = P(A) + P(Ac), so
P(Ac) = 1- P(A)  1.
• Theorem 1.3: Let S be the sample space of an
experiment. If S has N sample points that are all
equally likely to occur, then for every event A of S,
P(A) = N(A)/N,
where N(A) is the number of sample points in A.
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Shi-Chung Chang & Tzi-Dar Chiueh, Spring 2006
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• Remarks: There may be several possible sample
space for a particular experiment, Some of the
sample space may not have equally likely sample
points. See example in text.
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Shi-Chung Chang & Tzi-Dar Chiueh, Spring 2006
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