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**Unformatted text preview: **‘130 Deﬁne chapter 3 Probability Foundations Arr = Br: _ Brr+lr and observe that {An} is a countany inﬁnite collection of pairwise disjoint events (because the
{BR} are nested). Furthermore, ' 3,, = (11min) 0 (rigging = ujaﬂAJ-h 'F‘rom K4’, we have that P(B,,) = ZpgAj)‘. i2?! Because the inﬁnite sum we conclude that as desired to verify K4. ipmj) 5 1,
1 r . “E‘EOP‘BﬂﬂRXQHAIFO’ -
12 EXERCISES 153.1 133.2 _ E33 E3.4 (Subjective Probability) If you are currently taking a course in probability, we are innar- ested in how precisely you can estimate year subjective probability of the event G that your ﬁnal grade in this course will exceed your cnn'ent GPA. In order to do so, compare the likelihood of G to the likelihoods of events concerning the identity of the topmost card in a well-shuffled deck of playing cards. For example, is G less probable than the topmost card being either a heart, diamond, or club? Is G more probable than the topmost card being a nine? By considering-events that are more probable than G and ones that are less probable, determine the narrowest interval of probability in which you have sufﬁcient conﬁdence to bet accordingly with small sums“ I . (Relative Frequency) Select the Dow Jones data Set described in Problem E128. Turn it into a binary-valued data set by assigning a 1 if d,- z dJ-_1 and a 0 otherwise (is, a 1 if the price didnot decrease from that of the previous trading day). Choosing a range of years such as 1980-1989, does the relative frequency of ones appear to converge? ' an. Toss a penny 100 times and record the sequence of heads and tails“ How often did
heads occur? Plot the fraction of tirnes heads occur in the ﬁrst n tosses vs“ n, for n
a multiple of 5.‘ _ t _ 3 ‘ . by. Repeat this experiment only now Spinning the penny on .a tabletop. rather than tossing it“ Do the results on occurrences of heads seem to agree? , Select a thumbtack and toss it 100 times (be carefult), recording the sequence of outcomes as to whether the tack lands on its backfiread (B) (assign this case a l) or rests on its edge and point (E) (assign this case a 0)” Exercises E35 E3.6 133.8 n33 133.10 a. Informally, how predictable is the occurrence of the outcome B from your data
sequence? Can you think of a “simple” formula that predicts the location of the
next occurrence of B given the locations of. the previous commences? .- b. Evaluate the sample mean, median, and standard deviation for the resulting binary-- valued sequence.
c. Plot the fraction of times B occurs in the ﬁrst it tosses vs. n, for n a multiple of 5.. a. Prove that the relative frequency 7;, (A) is the following convex- combination: I r1304) é (1 131-104) + l-Inkvn)‘
R' ' I n b.. Provethat
_. 1
MA) --- Try-1M” 5 E 'c.. Does the fact that successive terms in the sequenceh'n (11)} necessarily converge to
each other, ' ' ' ' ' - ' ' "13,1130 lrn(A) - I'n—1(A)| = 0, imply that the sequence of relative frequenciesitself must converge? Let Q. = [0, 1] be the unit interval Define an event collection .A as the set of all events
that are ﬁnite unions of intervals. (The intervals can be of any of the four kinds.) a. Prove that A is an algebra of events. ' _ 1). Does A include the set R of rational numbers in the unit interval? c. Prove that .A-is not a a-algebta. -
Let Q = [0, H2 be the unit square. Deﬁne an event collection A as the set of allsubsets
of the unit square that are ﬁnite unions of rectangles. ' a. Prove that A is an algebra of. events. 3. ' . b. Is the interior T of the triangle with vertices (O, 0), (1/2, 1), (I, 0) a set in A? e: Prove that .A is not a a-algebra. ‘ '
Prof, Phynne has attempted to caICulate p = P(A) and has determined that it is a root of
the following. polynomial of degree ﬁve: ' ' ' (p — 2)(p ¥— ZJLTXP -+ 2J-Tf)(p + 5m; --. .5) .—..~ 0.
Can you help him choose an answer? I I If 9 = {(2, b, c}, the algebra A is the set of all subsets of Q, and the probability measure
P is partially dened by - PUG: = "5’ = "8! PG“: = "7, then complete the speciﬁcation of P for all remaining events. 7
Consider the event A: {(01, .. .. , mm}, where it is known that 0 < a 5'P({a>k}) z.- pk for k =1,2,. ...,,m, and that P(A) 5 b.
Show that b/a a m. " 132 £3.11 £3.12 E313
1513.14 133.15 _ £23.16 ‘ 133.17 E3.18 133.19 1513.20 Chapter 3 Probability Foundations If' {14,-} is a countably' inﬁnite partition of S2 and P(A,-) = abii a 0, then what are the ' constraints on a and b for P to be a probability measure? Show that (a) and (b) are incompatible with the Kolmogorov axioms. Construct a classical
probability example on a sample space of 10 points to show‘ that-(c) is compatible with b. Prove that a;
by
CS. a”
b. 9‘90?!“ 99"” the axioms“
' ‘ P(A-~‘B)=H7,P(B)'=.i4; .' Pol) = .‘6, 11(3) = n.5,P(A r13) 33.05; -
Pm) : ..6,P(B) = ..5,P(An B) 32,1391 UB) .—_ v.9“ Ii'P(A) .37, evaluate P(A“)i IfP(A) ..1 and P(B) = .2, what can you say about P(A UB) -}-P(A ()3)? If P(A) = ‘.2 and P03) = .‘4, what can you say about P(A UB)? If P(A n B) = ._1, and P(A U3) 5 n.5, how large can P(B) be? If PM 03) = .2, what is the largest possible value for the product P(A)P(B)?
Provide reasons for your answer: ._ .‘ Verify the following by appeal to the de Morgan laws and Kolmogorov axioms: P(A) +P(B) +P(A‘ nB“)=1+P(A Fl B)“ H H Is it always true that _
” P(A n3) _>_ P(A) +P(B) —_1? P(A)=IP(B_)=1=> Pot as) '.-= 1' ._
- and A its =>.P(A) 3 Pair)“ - If P(-A) = ‘;4, PG?) = ‘5, P(A n B) m Q3, evaluate P(A u B)“ If‘P(A nB) = 1, what can you say about the following: P(A), min(P(A), P(B))?
If'P(A UB) = 1, what can you say about the following: -P(A), max(P(A), P(B)),
and-min(P(A). P(B))?- - - - - .' - - - If'P(A) .2 .2, P03 «--A) = n.3, P(CnA“ GB“) :11, evaluate P-(A UB U C)‘
IfP(A) = ‘2, P03) 2 n.3,P(C) :-‘,1,AB = AC = BC 2 i5, evaluateP(A UB U C)‘. Is there a probability measure satisfying pm): n.1, 10(3) = ‘2, P(C) = .3, Potential: "‘7? $2={0, 1, .i.‘..,9},A={w:weven}.,B ={w:wodd_},andC = {1,2,3}. 3..
b., Can P(C) > max(P(A), P.(B))?
Can P(C) < min(P(A), P(B))? (If it is possible, provide- a probability assignment to the elements of $2 for which it is
true. If it is not possible, prove your claim.) '
E321 In a given random experiment 8, his known that Poo. 1, 2, 3}) = :.s, ‘P'({o. 1}) = “2” What can you conclude about P({0. l, 2}). and P({3})? Exercises - ' 133 E322 If we know that E323 E324 E325
E326 E3.27 E338 E329 . E330 E331
E332 PGOD = “2. Hill) = r-15. P({2}) m P ({3}) = ‘01. ‘ evaluate the probability that at least one of numbers 0,1,2,3 is the outcome of the
experiment. 5
Given three events deﬁned as the sets A1 = {09 2}: A2 = [1941‘ IA?! : {3! 5}, and having probabilities P(A1) = ‘3, P912) = ‘.2. P013) = ..4, what can you conclude
about each of the following probabilities? ' _a P(A§), P(A1UA2), Pm, UA3)..
bi. P({2}), P(At 0A3). '
o. P(U§=1A,-).r In Problem 131.18, the event Ak is that 1: clients are requesting service and we know that
k 5 3” We are given the probability assignment: ' i ' " ' ' P(At)_= .243, ' P(A2)=I.‘027,' P(A'3)f_-.;001‘.‘ a. What is the probability Polo) that no client is requesting—service? _
b.‘ ' What is the probability that at least two clients are requesting service?‘ it PM) = 3, Pas) = ..5, P(A DB) 2 ’what can you conclude about P-(Aﬂ B)? If $2 = {0, 1,2, .: ,9}, A = {0, l, 4},B = {3,.5},-P(A) = ..8,P(B) 2,1, what can you
conclude about the probability of C = {0; l,- 2, 3', 4}? _ ' ' ' " '
Repeat Example 3.? by determining the probability P(C) that the
Given three events deﬁned as the intervals“ Ar = i0, 2]. A2 = [1.4L A3 = [3,-‘53, and having probabilities P041): ..1, P(A2) = .2, P(A3) —.= n.3, that can you conclude
about each of the following probabilities? ' aw P045). POM U142): P(At U143)"
13.. P911 0A2), P041 0A3)”
Ct. P(U.3=1Ai)n Evaluate P(A LJB U C) when you know that
P(A) to ..3, P0?) = {.4, P(C) = “5,P(A n3) = “2, P(A n C) = v.1, MB n C) =j “15,P(A on n C) : “1., Two fair dice are tossed independently so that all ordered pairs of outcomes are equally
probable“ (See Section 2 .104.) What is the probability that the sum of the outcomes (number
of dots that are uppermost) is 11? ' If for-each i, P(A,-)-= .‘l, whatcan you conclude about P(LJ§=1A;)? : In a given message source that generates bytes, it is known that the probability of seeing
a consecutive “01” is v.05 no matter which of the positions 1, 2, .t ., w , ‘7 we examine for the outcome to is a multiple 135 ______“mb_* _m_ mm - __Chapter3 Probability‘Foundatlons
leading “0.” What can you say about the probability of at least one occurrence of .“01”
in a byte? E1133 Take 52 = [0, I], the unit interval, the event algebra .A = B as the usual Boolean algebra
generated by‘including all intervals in $2, and select the so-called uniform distribution 1
P(A)=f IA(x)dx.
' 0 Deﬁne the countable collection of disjoint events {A5}, as nonoverlapping intervals of
decreasing length, through .
1 d; I bi At = [ai,bil.ar = 5.51 = 1.ar+1= 3*.bi+1= 3"“ From this speciﬁcation, we observe that
1 . . . _
a,- = 531-‘,bi = 31-1, ti = bi --- a,- = aj,b.~+1 < a1". a. Evaluate the probability Hat). 4 ._
b. Evaluate the probability that at least one of the events {Ai} will occur. E334 In a photon-counting experiment,-it is known that the probability of observing exactly
k photons is e1“'1/k l,(where k! is factorial k) and this holds for all nonnegative integer
values 1:. Evaluate theprobability of observing an even numberof photons. (Hint: Consider the power series expansion fore" + 8“.) '
E335 There area. letters addressed to n ﬁshnet individuals and n appropriately uniquely
- addressed envelopes. The letters are randomly inserted in the envelopes. Let 3,. be the
_ event that at least one letter is in a correct envelope. 4 7 ' __ '
a. Exactly evaluate P(B,,) by uSing the lnclﬁsion-Exclusidn Principle:
b. Examine PUB") as n diverges. E136 Let :2 =' {am . .. .. , can} and take .A to be the power set of $2..
a. Verify/that for any to“ e 52 _ I
P“) = 154%“) satisﬁes the Kolmogorov axioms.
b. Can any probability measure P on .A be written as a convex-combination of the measures {P1, . .. . , Pu}, where
Fiﬁ) = JAG»)?
E337 Consider the target detection example of"Section 3.10 with the sole revision that . co 00
P904) m f dyIA(x. y)y‘xye'””"l’y .
o _ o . - ‘ . a. Evaluate the false-alarm probability when there is a single detector and the upper-
corner region D1 = Cab for a > 0, b > 0.. b. Use Boole’s Inequality to uppersbonnd the false—alarm probability Pm when there
are three detectors being used simultaneously and we claim detection if any of_ the detectors. do. Exercises . I _ 135 c. Use the Inclusion -—Exelusion Principle to evaluate PFA exactly in the case described
in (b). - 1313.38 Let s; = N ={0,1,. . ..} and
. _ 1 k P({m:w$k})==1— , for integer k 3 0.. ' ‘ ' 3. Evaluate PM”) for AFl = {w : a: > n}.
b.. Evaluate P(LJ.-;_.,,Aj).
(3. Do the events {An} occur inﬁnitely often with probability 1? ...

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