Chapter 11 Problems

Chapter 11 Problems - ‘ ~ 338 Chapter_11 Discrete...

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Unformatted text preview: ‘ ~ 338 Chapter _11 Discrete Conditional Probability 0 System Response (Goal 2) Sequence Probability Max:114.) = P(A1) I13 P941141. . . AH) Probability of Sequencelfiscade of Events A], .. .. .. , An 0 System Input from Output (Goal .3) Bayes’ Theorem _ - __ pm at.» 3-) - _ P(B:|A) -' kPmIBk)P(3k) ' Probability of Cause B.- GivenIEffect A (Inference) EXERCISES EILI. - "311.2 ' 1111.3 1511.4 EILS 311.6 , 1211.7 EILS E113 1311.10 ' - If 52 = {0, 1, .. .. .., 9}, p(0) = ..35,p(1) = .25, p(2) = -- » u = 31(9) = .05, A = {0, 4, 5},. and B = {2, 3, 5}, then evaluate 'P(A|B).. ' IfP(C ID) : .4 and P(DIC) = .5, which of C ,D is the more probable? If P(E) = .3 and P(F) ='-:7, what can you conclude 'ab’ou: P(E|F)? If {31,32} is a partition of Q,‘with P031) = .2, P(AIBI) m .5, and P(AI32) = .2, then ‘ evaluate P(31 IA). ' . _ Knowing nothing more, can you compare the values of P(X '5 Y]_Y = 2) and P_(X :5 SW = 2), and, if so, What is the comparison? (Explain Your contusion—do not give a yes/no "answer..) 7 ' ' _ Let PM) = 1/4, VP(B'V|A) = 1/2, and P(A|B) = 1/4.. a. Is A J. B"?! 5 ' b Does'A ‘eontain B or B contain A? 0. Evaluate P.(A“ lB“)._ - d. IS P(AIB) +P-(AIB‘) = 1? 8?. .15' HAIR) +P(A.‘lB) = 1? If P(A) >0, is it true that _ ' 'P(AnB:A)3P(AnBIAUB)? Event B corroborative of event A, denoted B 1i A, if‘P(AiB) > P(A).. Prove, or give a counterexample to, the following claims about being corroborative: a. symmettyzB “314:455 B. ‘ b. transitivity: A .3 B, B i; c =>A —‘*-;_c. c. 311A, CLAzss'nc 3A. Show that a. P(A[B n C) = P(A nBlC)/P(B|C); b. anthem) = P(A1|B) IT‘ nthAmtB n (0;? A»). 7 Statistics suggest that if it has been snowing in Ithaca, New. York, for exactly 1: days in a row, then the probability that it will not snow the next day'is #, k = 1, 2, .. .. ., . Find Exercises E11.11 311,12 E11413 Ell.l4 EILIS 339 the probability that a snowstorm that starts on January 25 will last through the end of the month, Message source M1 produces a word that is a byte, and allbytes are equally probable, Message source M2 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; '- a What is the probability that M2 produces a word that looks like a byte (i e.., no appearance of "2”)? . If a word is equally likely to come from either source, what is the probability that it will be a byte? _ _ .c, Given that we have observed a byte, how probable is it that it came from M1? Let B refer to inputs to a binary channel and A refer to outputs. The source is described by P(B_ = {0}) :14 The channel is described by - _, P(A.='{1}IB = {0}) = 1,2, P(A .—_ {me a {1}) = ,7, 3., Describe the outputs. a b, If we observeA = {1}, what is the probability that B = {'1} occurred, Message source M1 generates bytes having exactly 3 ones, with all arrangements being - equally probable. Message source M2 generates bytes having exactly 4 ones, with all arrangements being equally'probab'lel If we are equally likely to receive a message from M1 as from M2, what is the probability of receiving a byte with a one in' the first position? A message source is equally probable to be in any one of three states S1,S2, S3, In state St, the source emits a byte having no more than kbnes, withall such bytes being equally'probable. _ a. How many different bytes can this source emit? b.’ What is the probability that the source generates that the somce is in state 52? _ ' 1 ' ' ' e. What is-P(B)? d.. Given that B was observed, what is the probability that the source was in state Sg?‘ ' i * ‘ ' A signal S e {51, 32, .93} is the input to a noisy system having a nonnegative integer R b. the byte B 211000000, given as output]. The three possible Signals have respective probabilities of .2, .3, and .,5.J'I'he ' system is characterized by _ . 1311.16 - . . ' ' r 1 ' 1 1 _ ' Pia =us = so 3: (I —— final“. at; 5432.: 5.183 = 3.: a. What is the probability that the system output is 2? b, “If the output R z 2, then which is the most likely signal to and how likely is it? A particular circuit has ‘7 chips of type C1, 3 of type Cz, and 5 of' type C3. The proba- bility that a chip of type C,- will fail in the kth month of usage is ,.25i(1 —- 25f)", k r: ' 0, 1, 2, ,. ,., What is the probability that a randomly selected chip will fail in the second month? have been the input, ‘ 34 Chapter 11 Discrete Conditional Probability £111.17 A ternary-valued message with symbols drawn fi'om the alphabet {51, b. c} comes from one of the two message sources M1, M2. If the message source is M;, then the probabili- ties of the symbols being drawn are .3, .3, .4, while if the source is M2, the probabilities are .15, .25,- .25. respectively. The probability that source M1 is selected is .2. ' a. What is tlte probability of the message a? 7 . ' ‘ ' b. If we observe c, then what is the probability that it came from source M1? E11.18 A binary signalX with P(X = 1) _= .6 is transmitted through a BSC with error proba- ' ' ‘ bility of .1. I " " ' I ‘ ' a, Evaluate the probabilities of the possible outputs. I b. Determine the most probable signal to have been sent if a one is received, E11..19 An information source selects messages‘fr'om the set 8M??- {0, 1, 2} which are then transmitted over a memoryless nbisy channel that has the set of outputs Sc = in, b}. The channel is characterized by specifying the conditional probability P(C 1M ) for a channel output C, given a message M as input. We learn that no = alM = 0) = .2, PK: ; aiM _=,1)f=.14, P(C =a|M =2) = ' The information source is characterized by the probability 'P (M) that message i M is selected. We learn that ' ' ' ' ’ P(M = 0) = .3, P(M =._1). =‘ .2. a. How probable is it that we will receive C = b? H _ . b. If‘we receive C = b, then how probable is it that M = l was'sent? 13111.20 A binary symmetric channel (BSC) accepts an input X from the set {0, l} and produces ' an output Y in the saine The prohability of an error in transmission is p, no matter which error is made. An input is selected with probability P (X '=’ 0) = p0. ' a.' Evaluate-the probabilities for the possible outputs. - . _ b. Determine the probability that a one was sent, given that. a one was received at 7 the channeloutput. _ . _ E1121 On a multiple-choice question having at alternative answers, the prior probability that the student knows the answer is 'p. If the studenthas to. guess (event G), then all alternatives are equally probable. Find the probability that the student knew the answer to the question (event K) given-that he or she answered it correctly (event C ). 1311.22 A noisy cbunter' observes .nonnegative integer X and reports nonnegative integer Y such that ' - my exp; = x) = .75, PCY = x + 11X; x) ='..25.. We know that P(X =3) = ._s*+1 forx = 0, 1, .. . ’ ' ' a: Evaluate P(Y =2). _ , b. Evaluate P(X -_—— le =_— 2) for all nonnegative integers x. c. What is the most probable value ofX, given that we observe Y .-_- 2? 'Ir _ Exercises - 341 13111.23 A noisy counter observes X ~ PG) and reports nonnegative integer Y such that . “me—:3“ _ . i. Hr uk|X=;)— (kw)! (coir; >10“ a. Evaluate P (Y 7: 1).. by. Evaluate P{X = le = 1) for all nonnegative integer k(. c. What is the MAP estimator of X (most probable value of X), given that we observe Y = l? 7 E1134 An attempt is made to classify a sample of radioactive material by means'of a radiation ' count R over a fixed period of” time. It is initially known that the sample is either a rare element A with P(A) = .05 or a more commerr element B with P(B) = “95., We know that the probability mass function for R is 13(5) given that itis element A and 73(3) given that it is element 8.. -. - ' a, Evaluate P02 3 r)“ - - by If R .= 5 is observed, then what is the most probable source of the sample (i;.e., compare P(A|R = 5) with P031)? = 5)), and how likely is it to be that element? EILZS There are three urns U1, U2, and U3 with the following compositions of red, green, and blue balls: ' - . - ' ' rr=5.gr = 7ib1=0; r2=0,82'=10.bz=15: r3=5igs=5ib3=5r a.‘ If an um is selected at random and then a ball is chosen .at random from that urn, what is the probability P1, that a blue ball is chosen? _ b.. Given the previous information and that a blue ball was chosen, what is the probability Pg, 1' '= 1, 2, 3 that it was chosen from urn Ug,i = 1, 2, 3? c“ If, now, one ball is chosen at random from each of the three urns, what is the probability that two of the balls are blue? I I (1.. Under the conditions of part (c), what is the probability that the third ball is red, given that two of the chosen balls are blue? - ' E1126 a. For urn U3, what is the probability of choosing a second ball that is red, given _ that the that ball chosen was blue and that there is no replacement? bt. Repeat (a) for selection of balls with replacement. . E1137 A binary-valued Markov source produces a random sequence {Xi} with X.- e {0, l}, P(X1 = 0) = A, and (Vi > 1) P(Xi fixiiXi—r =-7Ci---1, "aXl =x1) = I a. Evaluate P(X2 = 1)., b” Evaluate P(X1 = 11X: = l), _ c., If this source is the input to a binary symmetric channel with outputs {Y.-} and ' error probability v.1, and errors are made independently on successive inputs, then evaluate P(Y2 = 1),. 342 ____ ____ _'_ _ _ _ _ _ _ _ _ w H _ _ _ mghapter 11 Discrete Conditions}! Probabilifl E1128 A binary erasure channel (BBC) has inputs X 6 {0,1} and outputs Y e {0, 1,E}., We know that P(X = 0) = .35 and that ' 'P(Y = OEX =0) = M7,-P(Y =E|X = 0) = 3, P(Y =OIX :1) =0, P(Y =EIX = 1) = “2‘. 3“ Evaluate the probability P(Y = E) of an erasure b.. Evaluate P(X = OIY = E), _ ._ c.. - What is the most probable channel input if we observe an erasure as output? E1129 Given that at least one. electron hasbecn emitted in I unsec, the probability of emitting exactly one electron is twice its unconditiOnal probability of being emitted in 1 insect Adopt an appropriate probability model. - . a, What is the'probabiiity of 0 electrons being emitted in 1 usec? b.‘ What is the probability of 0 electrons being emitted in 5 usec? c.. What is the average current? ...
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Chapter 11 Problems - ‘ ~ 338 Chapter_11 Discrete...

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