Applied Probability and Stochastic Process
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Transcript of Applied Probability and Stochastic Process
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ECE6111: APPLIED PROBABILITYAND STOCHASTIC PROCESSES
Yaakov Bar-Shalom
University of ConnecticutDept. of Elec. & Computer Engineering
Storrs, CT 06269-2157These viewgraphs are based on the text
A. Papoulis and S. U. PillaiProbability, Random Variables and Stochastic Processes,
McGraw-Hill, 2002and the viewgraphs of S. U. Pillai (www.mhhe.com/papoulis), with permission
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Basics
Probability theory deals with the study of random phenomena formodeling uncertainty.Probability demonstrates both our knowledge as well as our lackof knowledge.
Goal: to present methods that deal with uncertainty in engineeringsystems (some are inherently random, some we assume as randombecause of modeling limitations).Probabilistic Experiment
The notion of an experiment assumes a set of repeatableconditions that allow any number of identical repetitions.When a probabilistic experiment is performed under repeatableconditions, certain elementary outcomes i occur in different butcompletely uncertain (probabilistic) ways.We can assign a nonnegative number P (i) the probability ofthe event that outcome i occurs in various ways.
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Laplaces Classical Definition:The Probability of an event A is defined a priori without actualexperimentation as
P (A) =Number of outcomes favorable to ATotal number of possible outcomes (1)
provided all these outcomes are equally likely.
Example: Consider a box with n white and m red balls. In thiscase, there are two elementary outcomes: white ball or red ball.Probability of selecting a white ball is n
n+m .
Relative Frequency Definition:The probability of an event A is defined as
P (A) = limn
nAn
(2)
where nA is the number of occurrences of A and n is the totalnumber of trials.
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Axiomatic approach to probability (Kolmogorov):developed through a set of axioms (given below) is generallyrecognized as superior to the above definitions.
The totality of all outcomes i, known a priori, constitutes a set , theset of all experimental outcomes space of outcomes
= {1, 2, , i, } (3) has subsets A, B, C, etc., called events.If A is a subset of , then A implies .From A and B, we can generate other related subsets (events).
A B = {| A or B}
A B = {| A and B}
andA = {| / A} (4)
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A B A B A
A
A B A B AFig.1.1 Venn diagrams.
If A B = , the empty set, then A and B are said to bemutually exclusive (m.e.).A partition of is a collection of mutually exclusive andexhaustive (m.e.e.) subsets of such that
Ai Aj = andn
i=1
Ai = (5)
BA
1A
2A
nA
iA
jA
A B = Fig.1.2 The concepts of m.e. and m.e.e.
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De Morgans theoremA B = A B and A B = A B (6)
A B A B A BA B
A B A B A B A BFig.1.3
It is meaningful to talk about the subsets of as events, forwhich we want a mechanism to compute their probabilities.
Example 1.1:Consider the experiment where two coins are simultaneously tossed.The various elementary events are
1 = {H,H} , 2 = {H,T } , 3 = {T,H} , 4 = {T, T }
and = {1, 2, 3, 4}
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The subset A = {1, 2, 3} is the same as Heads has occurred atmost once" and qualifies as an event.Suppose two subsets A and B are both events, then consider thequestions
Does an outcome belong to A or B = AB = A+B?"Does an outcome belong to A and B = A B = AB?"Does an outcome fall outside A?"
Thus the sets A B, A B, A, B, etc., also qualify as events. Weshall formalize this using the notion of a Field.
Field: A collection of subsets of a nonempty set forms a field F if
i) Fii) If A F, then A F (7)iii) If A F, and B F, then A B F.
i.e., it is closed under set operations.
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Using (i) - (iii), it is easy to show that A B, A B, etc., alsobelong to F .
For example, from (ii) we have A F , B F and using (iii)this gives A B F ;Applying (ii) again we get A B = A B F , where wehave used De Morgans theorem in (6).
Thus if A F and B F then
F ={, A, B, A, B, A B, A B, A B,
} (8)From here onwards, we shall reserve the term event only to membersof F .Assuming that the probability (measure) pi = P (i) of elementaryoutcomes of are a priori defined, how does one obtain probabilities(measures) of more complicated events such as A, B, AB = A B,etc., above?The three axioms of probability defined next can be used to achievethis goal.
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The Axioms of ProbabilityFor any event A, we assign a number P (A), called the probability ofthe event A. This number satisfies the following three conditions thatare the axioms of probability.
Axiom 1 P (A) 0 (Probability is a nonnegative number)Axiom 2 P () = 1 (Probability of the set of outcomes is unity)(9)Axiom 3 If A B = , then P (A B) = P (A) + P (B).
The event is the sure event it occurs in every experiment.Note that (iii) states that if A and B are mutually exclusive (m.e.)events,
the probability of their union is the sum of their probabilities.
Q: What is missing in Axiom 3?
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The following conclusions follow from these axioms:a. Since A A = , we have using (ii)
P (A A) = P () = 1
But A A = , then, using (iii),P (A A) = P (A) + P (A) = 1 i.e., P (A) = 1 P (A) (10)
b. Similarly, for any A, A {} = {} (m.e.)Hence it follows that P (A {}) = P (A) + P ()But A {} = A, and thus P () = 0.
c. Suppose A and B are not mutually exclusive (m.e.)?
Q: How does one compute P (A B)?
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A BA A B
A B A BFig.1.4
To compute the above probability, we should re-express A B interms of m.e. sets so that we can make use of the probability axioms.From Fig.1.4 we have
A B = A AB (11)where A and AB are clearly m.e. events. Thus using axiom (iii)
P (A B) = P (A AB) = P (A) + P (AB) (12)To compute P (AB) we can express B as
B = B = B (A A)
= (B A) (B A) = BA BA (13)Thus
P (B) = P (BA) + P (BA) (14)110801 ECE6111L01 [Ch.1; Ch2] 11/ 29
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From (14),P (AB) = P (B) P (AB) (15)
and using (15) in (12)P (A B) = P (A) + P (B) P (AB) (16)
(This result really follows by inspection of Fig. 1.4 and compensatingfor double counting).
Question:Suppose every member of an infinite collection Ai of disjointsets is an event, then what can we say about their union
A =
i=1
Ai
i.e., suppose all Ai F , what about A? Does it belong to F?Further, if A also belongs to F (it is a measurable set), whatabout P (A)?
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The above questions involving an infinite number of sets can beunderstood using our intuitive experience from plausible experiments.For example, in a coin tossing experiment, where the same coin istossed indefinitely, define
A = heads eventually appears". (17)Is A an event?Let
An = {heads appears for the 1st time on the n-th toss}
=
t, t, t, , t
n1
, h
(18)
Clearly Ai Aj = . Thus, the above A is the infinite union
A = A1 A2 A3 Ai (19)
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We cannot use probability axiom (iii) to compute P(A), since theaxiom only deals with two (or a finite number) of m.e. events.To answer the questions whether A is an event and how tocompute P (A), extension of these notions must be done as newaxioms.
-Field (Borel Field) DefinitionA field F is a -field if in addition to the three conditions in (7), wehave the following:For every sequence Ai, i = 1, . . . ,, of pairwise disjoint eventsbelonging to F , their union also belongs to F , i.e.,
A =
i=1
Ai F (20)
This property is called being closed under a countable number ofunions. (countable = countably infinite)
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Thus, in view of (20), we can add another axiom to the set ofprobability axioms in (9).Axiom 4: If Ai, i = 1, 2, . . . are mutually exclusive, then
P
(i=1
Ai
)=i=1
P (Ai) (21)
Returning to the infinite coin tossing experiment, from experience weknow that if we keep tossing a coin, eventually, a heads must showup, i.e.,
P (A) = 1 (22)But A =
i=1 Ai and using the fourth probability axiom (21),
P (A) = P
(i=1
Ai
)=
i=1
P (Ai) (23)
Note: A is called a Borel set.
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From (18) An is exactly one in 2n outcomes. Thus, for a fair coin, wehave
P (An) =1
2nand P
(n=1
An
)=
n=1
1
2n= 1 (24)
which agrees with (22), thus justifying the intuitive reasonableness ofthe Axiom 4 in (21).In summary, the triplet (, F , P ), consisting of
a nonempty set of elementary eventsa -field F of subsets of a probability measure P on the sets in F subject to the fouraxioms ((9) and (21)),
form a probability model (probability space).The probability of more complicated events must follow from thisframework by deduction.
Q: What other events or variables might be of interest?
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Conditional Probability and Independence
In N independent trials of the same experiment, suppose NA, NB,NAB denote the number of times events A, B and AB occur,respectively. According to the frequency interpretation of probability,for large N
P (A) NAN
, P (B) NBN
, P (AB) NABN
(25)
Among the NA occurrences of A, only NAB of them are also foundamong the NB occurrences of B. Thus the ratio
NABNB
=NAB/N
NB/NP (AB)
P (B)(26)
is a measure of the event A occurrences given (conditioned) that Bhas already occurred.
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We denote this conditional probability by
P (A|B) = Probability of the event A given that B has occurred. (27)
Formal definition of the conditional probability
P (A|B) =P (AB)
P (B)(28)
provided P (B) 6= 0.
As we will show next, the above definition satisfies all the probabilityaxioms discussed earlier.
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We have
(i) P (A|B) = P (AB)(0)P (B)(>0) 0
(ii) P (|B) = P (B)P (B) =
P (B)P (B) = 1
(iii) Suppose A C = . Then
P (A C|B) =P ((A C) B)
P (B)=P (AB CB)
P (B)(29)
But AB BC = , hence P (AB BC) = P (AB) + P (BC).
P (A C|B) =P (AB)
P (B)+P (BC)
P (B)= P (A|B) + P (C|B) (30)
satisfying all probability axioms in (9).
Thus the conditional probability defined in (28) is a legitimateprobability measure.
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Properties of the Conditional Probability
a. If B A, then AB = B, and
P (A|B) =P (AB)
P (B)=P (B)
P (B)= 1 (31)
since if B A, then occurrence of B implies automaticoccurrence of the event A. As an example,A = {outcome is even} , B = {outcome is 2} , in a die rollingexperiment. Then B A and P (A|B) = 1.
b. If A B, then AB = A, and (if P (B) < 1)
P (A|B) =P (AB)
P (B)=P (A)
P (B)> P (A) (32)
For example, in a die rolling experiment, A = {outcome is 2} ,B = {outcome is even} , so that A B.The statement that B has occurred (outcome is even) makesthe odds for outcome is 2" greater than without that information.
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Total Probability Theorem (TPT)
We can use the TPT to express the probability of a complicated eventin terms of simpler" related events.
Let A1, A2, . . ., An be disjoint and their union is they are m.e.e..Thus AiAj = , and
ni=1
Ai = (33)
Thus
B = B(A1 A2 An) = BA1 BA2 BAn (34)But Ai Aj = BAi BAj = , so that from (34)
P (B) =
ni=1
P (BAi) =
ni=1
P (B|Ai)P (Ai) (35)
The TPT is an extremely useful result the key is in finding thesimpler conditional events.
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With the notion of conditional probability, next we introduce the notionof independence of events.
Independence:A and B are said to be independent events, if
P (AB) = P (A)P (B) (36)Notice that the above definition is a probabilistic statement, not aset theoretic notion such as mutual exclusiveness.
If A and B are independent, then
P (A|B) =P (AB)
P (B)=P (A)P (B)
P (B)= P (A) (37)
Thus if A and B are independent, the fact that B has occurreddoes not shed any more light on the event A. It makes nodifference to A whether B has occurred or not.
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An example will illustrate the concept of independence.
Example 1.2: A box contains 6 white and 4 black balls. Remove twoballs at random without replacement. What is the probability that thefirst one is white and the second one is black?
Let W1 = first ball removed is white"B2 = second ball removed is black"
We need P (W1 B2).We have W1 B2 =W1B2. Using the conditional probabilitydefinition,
P (W1B2) = P (B2|W1)P (W1) (38)=
4
5 + 4
6
6 + 4=
4
15
Q: Are the events W1 and B2 independent?
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Are the events W1 and B2 independent?To verify this we need to compute the unconditional probability P (B2).
The first ball has two options: W1 = first ball is white or B1= first ball isblack.Note that W1 B1 = and W1 B1 = 1. Hence W1 and B1 arem.e.e., i.e., they form a partition of 1. Thus, with the TPT,
P (B2) = P (B2|W1)P (W1) + P (B2|B1)P (B1)
=4
5 + 4
6
6 + 4+
3
6 + 3
4
6 + 4=
2
5
and
P (B2)P (W1) =2
53
56= P (B2W1) =
4
15
As expected, the events W1 and B2 are dependent because theremoval of the first ball, which is not replaced, affects the probabilityof the second removal outcome.
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Bayes Theorem
From (28),P (AB) = P (A|B)P (B) (39)
Similarly, from (28)
P (B|A) =P (AB)
P (A)or P (AB) = P (B|A)P (A) (40)
From (39) and (40), we getP (A|B)P (B) = P (B|A)P (A)
or
P (A|B) =P (B|A)P (A)
P (B)(41)
Equation (41) is known as Bayes theorem or formula (or rule).
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Interpretation of Bayes theorem
P (A) represents the a priori (prior) probability of the event A.Suppose B has occurred, and assume that A and B are notindependent.Bayes formula in (41) takes into account the new information (B hasoccurred) and yields the a posteriori (posterior) probability of A givenB.
A more general version of Bayes theorem:Let Ai, i = 1, . . . , n represent a set of m.e.e. events with associated apriori probabilities P (Ai), i = 1, . . . , n. With the new information B hasoccurred, the information about Ai can be updated as follows
P (Ai|B) =P (B|Ai)P (Ai)
P (B)=
P (B|Ai)P (Ai)nj=1 P (B|Aj)P (Aj)
(42)
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Example 1.3:Two boxes B1 and B2 contain 100 and 200 light bulbs, respectively.The first box (B1) has 15 defective bulbs and the second 5. Suppose1. a box is selected at random and 2. one bulb is picked out.(a) What is the probability that it is defective?
Solution: Note that box B1 has 85 good and 15 defective bulbs.Similarly box B2 has 195 good and 5 defective bulbs.Let D = Defective bulb is picked out. Then
P (D|B1) =15
100= 0.15, P (D|B2) =
5
200= 0.025
Since a box is selected at random, they are equally likely, i.e.,
P (B1) = P (B2) =1
2
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Thus B1 and B2 form a partition of (part 1 of the experiment), andusing the TPT (35) we obtain
P (D) = P (D|B1)P (B1) + P (D|B2)P (B2)
= 0.151
2+ 0.025
1
2= 0.0875
Thus, there is about 9% probability that a bulb picked at random isdefective.(b) Suppose we test the bulb and it is found to be defective. What is
the probability that it came from box 1, i.e., P (B1|D)?
P (B1|D) =P (D|B1)P (B1)
P (D)=
0.15 1/2
0.0875= 0.8571
Notice that initially P (B1) = 0.5; then we picked out a box atrandom and picked a bulb that turned out to be defective. Canthis information shed some light about the fact that we mighthave picked box 1?
From the answer, P (B1|D) = 0.857 > 0.5, and indeed it is more likelyat this point that we must have chosen box 1 as opposed to box 2.Q: WHY?
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Independence of n events A1, . . . , AnP (Ai1 . . . Aik) = P (Ai1 ) P (Aik) for all sets of integersi1, . . . , ik n.Pairwise independence of n events complete mutualindependence?P (AiAj) = P (Ai)P (Aj) / A1, . . . , An independent.Example:
A
B
C
P (A) = P (B) = P (C) = 1/5P (AB) = P (AC) = P (BC) = P (ABC) = 1/25
P (ABC) 6= P (A)P (B)P (C) = 1/125 = NOT independent !!
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