Chapter 02 Problems - Exercises 89 EXERCISES E11 132.2 E2.4...

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Unformatted text preview: Exercises 89. EXERCISES E11 132.2 E2.4 Many good problems can be found in the classic text by Feller' {271, although they are not in the language of. electrical engineering. (Scrabble is not random.) In what follows, we refer to the usual 26 alphabetic characters of English and restrict vowels to a, e, i, o, u. ' ' a. How many words are there of length 2 without regard to the constraints of English? b In Scrabble (as of March 2,1998), there are 96 acceptable two-letter words What is the probability that a randomly selected pair of letters produces a Scrabble word? c. In Scrabble, there are 37 three-letter words that begin with the letter c What IS the probability that a randomly selected two-letter continuation of c will produce a Scrabble word? . d.‘ In Scrabble, there are 62 acceptable four-letter words containing exactly three vowels. What is the conditional probability that'a randomly selected four-loner word will be a Scrabble word given that it has exactly three voWels‘? . How many distinct words (not necessarily meaningful ones) can you form from the following, using all of the given letters: a. The author 3 first name TERRENCE? b.. The author’s last name FINE? c. Your first name? d. Your last name? - - e. The word PROBABILITY? Message source M produces a word of length eight characters, with the characters drawn from the ternary alphabet {0,1,2}, and all such words are equally probable What is the probability that M produces a word that looks like a byte (i.e.., no appearance of ‘2’)? A ternary communications system sends packets composed of characters drawn from the alphabet {0,1,2}. a If a packet has nine characters, how many possible packets are there? b If all possible packets are equally probable, then what rs the probability of a packet that does not contain the character ‘2'. 9 E215 E216 E2..7 E2.8 E219 112.10 132.11 Chapter 2 Classical Probability c What is the probability of a packet containing equal numbers of each of the characters? A signal transmission path has five links that are serially/cascade connected, each of which can be 1n any one of 10 distinct states a. How many distinct transmission. paths can there be (assuming that the sequenccfortler of the links matters)? b. What' rs the probability of such a randomly chosen path having the second link 1n _ state “1”? c.. How many paths can there be if we use three special links in place of the first three of the original links. and these special links can be in any one of 20 states? Consider the following simplified packet routing problem: A packet is sent to a first router and this router uses 2 bytes to address a second router. The selected second router then uses-a l-byte address to send'the packet to the end user. , ' _a.. How many different second routers can the first router reach? 111 How many and users can this system send a packet to? In QPSK, there are four possible signals {410,131, .92, 93}. What is the minimum length n of messages (sequence of QPSK signals) needed to ensure that there are at least 1,000 different messages? 1, _ _ _ . . . . Given the integers 1; 11.1.1, 11, how many subsets of size It 5 'n have their largest member the integer r‘ < n? What 1s the probability of the event A that when a k-tuple of integers is drawn at random with replacement from 1,. ,,n the largest integer chosen is 1‘. ‘7 Verify the Format combinatorial identity 0 21) a. How many k--tup1es of positive integers x1, 1. .. 1,1:1‘ satisfy Zizk-i-l? i=1 b1. Verify that there are (3:11) _k—tuples_ of positive integers x1, 1. 1. 1,1511 satisfying . .1 . 211:1. i=1 'E2..12 What is the probability that a randomly selected 64-bit register sequence will start with 132.13 00 and have exactly four zeros? . a. Let . 1‘}, denote the number of {0, 1}—valued binary sequences of length n in which there are no adjacent ones. {Codes for magnetic recording have constraints of this type') Evaluate fbtflr afldfz ‘/’\ Exercises 132.14 132.15 _ E216 E2.1‘7 E118 E2..19 91 b._ Verify the Fibonacci recmsion a =fia—1 +1.-.. _ for n z 2. _ . c. If all binary sequence 1:}, ...,.x,, of length n have equal probability, evaluate the . probability 11,, that we observe a'sequence having no adjacent ones. A. message source generates equally probable messages of length 10 from a symbol alphabet {— l, 0, 1}, satisfying the constraint that no two consecutive symbols are the same in any message (e. g. cannot have- --00-- 1- --). a. How many possible messages can this source generate that satisfy the constraint? b. How many possible sequences of length 10 are eliminated by the constraint? c. What is the probability of a message beginning with the sequence of symbols --1, l? d What is the probability of a message having -—-1 in the fourth position? A random byte rs received consisting of eight bits b1,.. ,b3, b 6 {0 1}. a. If all bit patterns are equally probable, what 1s the probability of receiving 101010109 b. What rs the probability that b1.— -— l9 c..What'1s the probability that all of the bits have value 19. d What re the probability that at least one bit has the value 19 e. What is the probability that In +b2 = 19 .. ' How many binary strings of length 211 have r1 ones in the first half and 'r2 ones in the- second half? Consider the 30 leading digits in the first two columns of Gaussian random number data listed in Section 1.11. We hypothesize that these digits are drawn at random from —9, —8. .,,0 '1,. 9. It rs clear from the presence of duplications that these draws must be wiih replacement a- If this is the case, then what rs the probability of observing as many of —l as we did? b. Is the hypothesis we made of random draWs supported by. the probability of this event? a. What is the probability of drawing five cards all different, from the usual well-- . shuffled deck of 52 cards? b. Repeat the calculation if the five cards are new drawn with replacement. c. What' rs the probability of drawing five cards (without replacement) having the same suit? At an- IEEE Banquet there were 13 faculty and 65 student attendees. Seven tee shirts were awarded by random drawing without replacement. a. What rs the probability (evaluated by classical probabifity) that the first two indi- viduals selected were faculty? b. Provide an expression for the probability (evaluated by classical probability) that the students receive no more than three tee shirts (i. e. no more than three students among the first seven individuals selected) . 92 Chapterz Classical Probability 132.20 a If a language has 200 syllables, how many words can you form having no more than three syllables? b. How many binary sequences are there oflength 100 with exactly 40% ones? E221 A message source generates binary messages of length 1,000. Each message has exactly 300 ones, and _all such messages are equally probable .What rs the probability of a message with all ones in the initial 300 positions? __b. W‘What is the probability of _a message with all bass in the first 600 positions? c. If all possible messages are to be compressed to the same length 111, how small can at be? What rs the bit- -per symbol entropy of this message source? How many undirected labeled graphs are there on 10 nodes? How many undirected labeled graphs on.10 nodes have five links? For what number m“ of links do we have the largest number of' undirected graphs on _11 nodes? - E2.23 If possible, sketch a graph on six nodes having the degree sequence 3,2,2,2, 1,0. , £22.24 What' rs the probability that a randomly. selected undirected G4 will be connected? ' E225 Is it more probable that a randomly selected. undirected 65,- will be connected than that a randomly selected undirected 65 will be connected? . E226 a. How probable is it that a randomly selected. undirected graph on six nodes will have exactly 10 links? ' _b.. How probable' is it that a randomly selected directed graph on six nodes will have exactly 10 links? 112.22 99‘!” 9- ._ E2. 27 A communications network or bulk power transfer grid is described by the graph of _..Figure 2.5 m which an. edge or straight-line segment denotes a communications link or power transmission line and a numbered box' is a node or vertex and denotes a switch or receiver of communicatibns or a load or switching substation. The links will be assumed to be directed 1n that a message or power can only pass born a lower numbered node to a higher numbered node. a. Verify. that there are exactly three directed paths from node 1 to node 6 b. How many possible directed paths are there from node 1 to node 13? c. How many possible directed paths are there from node 1 to node 14 if the path must contain node 5? E2 .28 For the directed graph depicted' in Figure 2.5 and the conventions about links directed from lower numbered nodes to higher numbered nodes, evaluate the indegree and outdegree sequences. E229 For the directed graph of Figure 20. 3 list the sequences of outdegrees and of indegrees E230 a. For the WWW, how many more pages are there with 10 links pointing to them than with 100‘? b. How many mere Web pages are there that point to 10 other pages than that point to 100? -' E231 A message sourCe M generates equally probable messages of length 10, from a quater- " nary symbol alphabet {12'9ka 0, l, 2, 3}, that must satisfy the constraint that no two imaginary symbols can be adjacent. 3’: .f‘“\.‘ 93 Exercises _ . -- . . . E232 122.33 E234 E235 £2.36 Figure 2.5 Communications or Power Grid Network Graph. a, How many possible message'sequences can M generate satisfying the constraint? b. How many messages are eliminated by the Constraint that no two imaginary symbols ' ‘ can be adjacent? c. 'What' rs the probability of a message beginning with the symbols" 1 - i? d. What is the probability of a message having- -'1 m the fifth position? Consider a message source 8 that uses a tiernary alphabet {a, b c} to generate a message of n > 2 symbols. 8 satisfies the additional constraint that the first and last symbols m a message must be the same. All messages satisfying this constraint are equally probable. What' rs the probability that a message contains exactly tWo occurrences of the Symbol a? A message source 8, using the quaternary alphabet {a, b, c, d}, produces equally probable messages of length 8 that satisfy the constraint that each message must contain exactly two uses of the symbol a a. How many different possible messages at can 8 emit? .' b. What 1s the probability of event A of a message having its first two symbols be an? e. Given that the message starts with aa, what rs the probability of event B that the last symbol' rs a b? (1.. Given event C that the message starts with ab, what rs the probability of event D that the last symbol" is a? Professor Phynne believes that 100! rs larger than 100”. Use Stirling’ s formula to deter- mine whether he is correct. (Hint: e23~ N 10.) What' rs the probability that there will be exactly one matching pair of birthdays in a group of r > 2 people when the number of birthdays n- _ 365 and We assume that all birthdays are equally probable? In mobile cellular communications systems, a slotted Aloha protocol is need to resolve collisions between callers seeking to place a call. in the same time slot If two callers ' “collide” (attempt to call in the same time slot), then a simplified form of the protocol-is 94 Chapter 2 Classical Probability _u..-....__._.-_.....__.a._...-..m.—.—..— 1—..-"—..__..._———_._.__...._._ E237 E238 E239 - 112.40 132.41 that caller 6; attempts to place her call at a randomly selected time slot t,- 6 N19 and that all pairs of time slots 131,12 are equally probable. a. Given that Cl and c2 have collided, what is the probability that caller c1 will attempt to call at time slot 1‘; = 2 and caller c; attempt to call at time slot 1‘2 = 9? b. What is the probability that-two callers will succeed in placing their calls (t1 tyé 12)? In a mobile cellular telephone system, the service region is divided into c distinct cells, each serviced by a base station. Assume that, at a given time, there are u > 0 distinct clients and that all allocations of clients to cells are equally probable. A base station in a cell can handle at mostf clients, Wheref < u <‘cf. a. How many distinct allocations of clients "are there? b. What is the probability that Cell 1 cannot serve all of its clients? c. Provide an expression for the probability that all clients can. be provided with service; that is, 110 cell has more than If clients. a. How many texts of length 52 are there that use each of the 26 letters of the alphabet exactly twice? b. What is the probability that, in a text generated as in (a), you will observe the consecutive occurrence dd? A radar partitions its surveillance range set 72 into nonoverlapping range gate sets [61, .... G.,3.} At a particular time, there are exactly a aircraft in 1?. Assume that all - arrangements of aircr'aft' 1n range gate sets are equally probable a. ._ What rs the probability 1",}l that no two aircraft share the same gate set? b.‘ What is the probability Pb that at least two aircraft share the same gate set? c. What' _1_s the probability 1’ch) of exactly 1: aircraft 1n gate set 61? (1 Show, using the Binomial Theorem, that Pc (k) sums over It to unity, as it should. A device has E > 1 energy levels and N > E free electrons, with any number of electrons allowed to occupy each energy level. .a. How many distinguishable energy level occupancy arrangements are there? b. If all such arrangements are-coually probable, what is the probability Pb that all of the-energy levels will be filled by at least one electron? .c..- Verify that 0 5'P1,__ < 1. ' 1 ' k photons are known _to have arrived in the time interval [0, N), which we take to be partitioned into unit— length intervals, [0, N) .— U?’_1[i —1 i). All arrangements, of the k indistinguishable photOns in the N (> 1:) given intervals of unit length are equally .. - probable. 132.42 a. What is the probability Pa that no two photons occupy any of the unit length . . intervals of the given partition? . b. What 1s the probability P}, that at least two photons share some unit length interval? c.. What. rs the probability PC of exactly two photons' in [0,1)? In a given experiment, we observe three photons occupying some of 10 cells {q1. , , gm} in quantum phase space. - . - Exercises 152.43 1132.44 E2.45 £2.46 E147 132.48 ___________m__.m__._._____.__.-._____.______.._.._2_5. a, How many physically distinguishable arrangements 11: are there for placing the _'photons in the cells? - b. If all of these arrangements are equally probable, what 15 the probability of event A that all three photons are in cell q]? c. What is the probability of event B that none of the photons are in cell 43;? In a given experiment, we observe three electrons satisfying Fermi—Dirac statistics occu- pying some of 10 cells {(11, i. .,qlo} in quantum phase space a. How many physically distinguishable arrangements at are there for placing the electrons' in the cells? , b. If all of these arrangements are equally probable, What is the probability of event A that there is an electron in each of cells 91, :14, qg? o. What is the probability of event B that none of the electrons are in cell ql? 1 - A message soiirce 8, using the quaternary alphabet {01, b, c, d}, produces equally probable messages of length 8 that satisfy the constraint that each message must contain exactly three uses of the symbol a, a. How many different possible messages m can 8 emit? b. What is the probability of eventA of a message having its first two symbols be aa? 1:. How many messages 11 are there in which a and I: appear exactly once? An image is composed of a 20 x 20 array of pixels that are either black or white a. How many possible images are there? ' - b.. If an image is chosen at random, what is the probability of its having exactly 100 black pixels? c., What IS the midst probable number of white pixels? ._a.J How many ways are there to arrange 1 students' 1n a lecture room with 11 seats? b If r = 130 and 11- ._ --,200 evaluate your answer, using Stirling’ s formula A 10 x 10 pixel image is composed of 20 red pixels, 30 green pixels, and the remainder blue pixels at, How many such images can you form? - I b.. In such a randomly generated” 1mage, what' IS the probability that the first pixel 15 red? ' c, If an image is selected at random, what is the probability that the first row will contain no blue pixels? .d“ What' is the probability that the first pixel is red and the last pixel 1s green? 'An image of a character is represented by a 5 x- Tpixel array, with each pixel taking on one of four possible gray-levels 31, . .. 2. , g4..- ' a How many possible image representations are there? ' b. What' is the probability that a randomly selected representation will have no pixels - containing level g4? c, What 15 the probability that a randomly selected representation will have a first row of pixels all at level 31? 96 . 1:32.49 E250 E251 ‘I _Chapter 2 Classical Probability. In a given classical probability problem it is deteimined that the events {31, 32, 33} are pairwise disjoint and that their union is the sample space. These events have the following probabilities: 13031) = .2, P032) = 3 There. is another event A about which it is known that P(A|BI) ..—.. .3, P041132) = ..4,P(AI33) = .1. :1. Evaluate P(A).. b. Evaluate P(leA).., Is it always one that, for P0?) > 0, P___(__A)., A P( lB)< _ P (B) You must give reasons for your. answer. In a given communication system, all bytes are equally probable. Define the weight w of abytebtobe- ' ., ‘. -' 8 = [111.122. webs]. b.- e {0.1}. w =21».- - 1 ' Define the following events: 112.52 E253 ' E254 E255 A =_{bi:b1 z b, 4.91}, a = {P : w is odd}, ' Evaluate P(A), P(B), P(B IA), and P(AlB). . a. If we toss two well- balanced dice and observe the event A that the sum of spots showing uppetmost is 9, then what' is the conditional probability 'of the event B that the first die is showing 4 Spots? I). Are the events A and B independent? " a. If all bytes are et1ually probable, then what is the probability of Obsetving one containing exactly 3 ones? b. If we observe exactly 3 ones, what is the probability that the first bit' is a One? - c.A1e these two events independent? In information theory, we are interested in so-called typical sequences: sequences having a -'spec1fied composition. In this problem, we consider sequenbes of length 30, with elements drawn from {0,1,2}..All such sequences are equally probable. a What' is the probability that a sequence will be composed of. elements that are just ones and twos? -b. What' IS the probability that a sequence will have 5 zeros, 10 ones, and 15 twos? . c.. What' 1s the probability that a sequence will have zeros in its first 5 positions, given that it has the composition specified' 1n part (b)? . We observe an X = S +N that' is the sum of a signal S taking values in {0, 1} and a noise N taking values in l— -1, 0, l}._ Assume that all possible pairs (shay) of values 5,- for S and n, for N are equally probable. a. What' is the probability pk: P=(X k) for each value 1: taken on by X. 9 b. Evaluate P(S = 0).. F“. 97 Exercises . 0.. Evaluate P(S m (51X = k) for each value k taken on by X .. d. Are S = 0 and X m —1 independent? E256 We observe Z = XY, where X e {0, 1} and Y E {—1, 0, 1}. Assume that all possible pairs (xhyj) of values x,- for X and yj for Y are equally probable. a. What is the probability pk = P(Z = k) for all possible values 1: taken by Z? b. Evaluate P(Y = 0).. 0. Evaluate [’0’ = 012 = k) for all possible values 1: lo: Z. E257 A woman and a man (unrelated to each other) each have two children. Assume that all four poSsible arrangements of “boy” and “girl” for the pair of younger and older child, 5?), bg, gb, gg, are equally probable. At least one of the woman’s children is a boy, and the man’s older child is a boy. (Hint: use Bayes.) a. Show that ...
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