Chapter 12 Problems - 370 Chapter 12 Mixed Conditional...

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Unformatted text preview: 370 Chapter 12 Mixed Conditional Probabillty ‘fi’lXO’IxflPXWZH f? (Y) 7 ' ' IPY=iX= X, Y ='-“ C, D: .fx'y(x'lyi) = ( Z—l xJfi._._(x). ” X,Y 'j—'-'D,C: P(X=leY=y) PY 0ft) EXERCISES E12.1 ' E122 E123 E12.4 1312.5- E116 E125 E123 If fi;_y(x; y) =' 4xy on {0, I]2 (the unit square) and 0 otherwise, then evaluatefylx for 0 < x < l.‘ For appropriately chosen constant 0, Jim (x, y) = ce“"" "1” U (IOU (3’)- Evaluate P(X 5 3]Y a 2)" A system .S‘ has input X and output Y“ a If the pdfsfk (x) and .fy (y) are known, then what can you say about 8? b.‘ If, instead, you are given fyng‘Lt) and fxly (x 13:), what can you say about fl; and ff? A stochastic system with inputX ~ .N'(0, 1) has response Y characterized by‘fymy Ix) = 1 '- — - - . is l” "H Provrde expressions for fr, fiqy“ A system .S‘ transforms X into Y such that Le" %(x2‘+}l‘2) " mom) == 2” Describe 3.. What is known about Y? ' Given that we observe Y = 3;, what is known about X? rtfmozix) = Ix +1|e-L"+'W U0) and X ~ 3(2, g), what'can you say about Y? b.. What is known about X given 1’ = y for y > 0? ' A component is purchased from Company A or from Company B, each with equal probability. If it comes from A, then the probability that it will not fail prior to time _ t equals e"‘U(t); if it comes from B, then the probability that it will not fail prior to fimmtke"mU6) _ " u- a‘. Express the probability. that the component will fail prior to to” b‘. It was observed that the component failed at some time in (£1, 1‘2).._.Find the probability that it was manufaetmed by A. " . An analog voltmeter can be thought of as providing a reading X that is the sum of the true voltage V' and an unlinked (independent) noise N- that is distributed as .N(0. 02).. In this case, fxlflxlv) is .N'(v, oz)” If the true voltage V is provided by a random source distributed as .N'(m; 1), then calculate I fx,‘ fiqx” (Hint: The answers will again be normal.) 5” 997:” Exercises E129 1512.10 1312.11 E1212 E1213 371 A signal X ~ U(-1, 1) is transmitted through a noisy channel and received as Y, with I Lye—2x)? y; a. Describe the response Y by providing an expression forfy. b. Given that weobserve Y = .5, what value of X .is'the most likely (i.e.., has the highest conditional density)? frlXO’lx)= If 37if05x51, OSny2 I x , z a fxfl ’0 {0 otherwise then determine fylx (ylx) for 0 < x < 1, taking care to specify-your answer for all values of the argument 3:. _ a. Show how to determine fy from fymfigly. b. If it is claimed that ' I 1 _ __ 2 .leXU’ix) = —e (x i) , .12 - I 'J—8"17-La file(xIY)=fi J2? then can we determine fy? - A system 8 has input X and output Y with joint density me,» = chwo —-x)U<1+-x wane—'1‘- a. Describe 8 through I fylx. If we observe Y = 2, what do we know about X? An amplitude-modulated (AM) signal has random phase angle (9 e [—rt, 1:] and random - amplitude A a 0 having .a joint density 1312.14 £12.15 ' , l fonflia) =. :3- Evaluate the cenditional density fem. _ . . ‘ We observe a voltage V that arises from noisy reception of a signal S drawn from {1, 2}. The voltage V has the conditional density fi/tsMS') = (s + 1)v“U(v)U(1 -- v), and the prior signal probabilities are given by P(S == 1_) = 1/3. The MAP (maximum a posteriori) signal estimation rule requires us to determine P(S = s [V = v) and then select that value s" which maximizes this term. ' ' - a. If we observe that V = 1/2, then what is the MAP estimate of'S? b.. Futthetmore, how probable is it that this estimate is correct? - A random systems has input X and output Y related through "“2U(a)U(6 + more: -— a). . 1 _ ‘- Qfi’iX(YIx)=-§e 1’ ‘1“ a. IfX ~ 11(0, 1), then describe the system output in the regime y > 1.. b If we observe that Y = 2, then describe the system input. 372 ' Chapter 12 Mixed Conditiona! Probability 1 _ was 28 fylx (Y [15} = is a pdf b‘ If X ~M(0,l)eva1uatefiu. 13312.17 A system S with inputX and output Y is such that km y)=-{% ifx2+y2 <1 _ 0 otheiwise 2. Describe the input X .. - b‘. Describe the system 8 for lxi < 1.‘ » , c.. If we observe Y = 0.5, then what do we know about X? 1312.18 X ~ not) is input to a system 3 with output Y described by . ' 3 _1 ‘ — 'f'ks 5k 2 fYJxUlk)=§_U0"~~'k)U(k+2-y)= 2 ’ y + a. . _ 0 if}: other a. Describe-Y“ - bi. Describe what is known about X if we obsexve Y m 3.5“ 13.12.19 X IMO, I) .is input to a system S with nonnegative integer output Y a. 8 is described by PC? =le =x)=(_1---x)x”-‘. an Describe the output Y . _ b. Describe what is known about the input X, given that we observe the output Y = 2‘. E12120 We have I ‘- FX.Y,Z (3733?. Z) n“: miter“: + 6—39 + e_z].._] H at. Evaluate fig; 2 through the triple partial differentiation of Emu" obi. Evaluate the cdfolflx, y)” c.. Evaluatefzmi/(zlxmi- ' E1231 A signal X ~ 5(1) is transmitted optically by encoding it so that the number K of received photons has expected value X _ - a. What is P(K = kiX =x)? ' b., Show that P(K at) is rug)" _ ct Given that we have received K = 2 photons, what do we know about X? E1222 A system noisily reconstxucts an analog signal S from a digital signal D.. It is known that P(D :11): is“, d = o, 1, .n .n fs'IDGM) -_- geek-41., - Exercises ~ ' ' a. Describe the analog signal S i. b. If we observe S = 5, what do we know about D? c.. Given that we observed S = s' < 0, what is the most probable value ofD? E12.23 A noisy 3-bit AID converter accepts an input S with fs-(s) = 3.S'_2U (.s_)U(l — s) and generates an output D ~ BU, .S')“ a. Evaluate the probability description of D and ED; b.‘ What does observation of D = d tell us about S = s? - c.‘ Given that we have observed D =' 2, what is the most likely value of the original - signal S? 7 £12.24 A system S has input X- and-output Y, and we know fx y‘. at. Describe S.‘ b. Describe what is known about the input When we observe the output“ 1312.25 a“ Given fnx,fi(, what can you say about fy? r _ ' bi. Given fylx,fy.‘What can you say about 15‘? - - _E12.26 The state X of a finite-state system is measured by a noisy analog measurement Y, with the measurement related to the state by Y IX ~ 8 (X ). (Y conditional upon X = a is 8(a).) You know that each of the four possible state values 1, 2,3. 4 is equally probable [0 occur; . a. Describe Y‘. H . _ bi. Describe what you know about X, given that we have observed Y H c.. What is the most probable value of the state X, given that Y = 1.5? E1237 _ We observe K ~ 802, Jr) with 7: unknown“ If 71' is chosen to be fi(m, n), then evaluate whatK=ktellsyouaboutirsp‘u... _ . -I _ . . E1228 We obsexve a measurement Y oi'X, where we know that the conditional density of“ Y given X . = x is laplacian'with parameter at (x) = x . We are told that X ~ B(n, 21-)” What ' does_Y=ytellyouaboutX=.-_'x? _ ' ' ' _ ' E1239 YOu are giventhat the state vector X N N(0, Q), you observe Z = + V with V ~ .N (0, R), and the matrices are specified by I ' -_ _1"o"_1'1'fl_10. ,Q-(o 2): lat—(1 «-1)’ 1R—(o -. What does Z =: z tell-you about X = x? ' ' ' - = £12.30 Consider X with EX = m = (1, 2. 3)T and positive definite covariance matrix C that is randomly chosen as follows: Using Matlab, -. ' , - ‘ - ' _ _ n=zanan(3,3), C=A7A+0011L I Treating the selected C as nonrandom and assuming that va Nun, C), eitplicitiy calcu- . late the conditional density flaw] X2. - r...
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