Chapter 08 Problems - Exercises“ 1 5 247'I‘lteisecond...

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Unformatted text preview: Exercises“ ' -- -- 1 5 . 247 'I‘lteisecond case is that 1 2;- 3’1 ”:3 #2,? " " 1" 3 13301;) 2“ The denizity f“ is G for 5 set) 311' ’for m. _... b. uu._.._.‘-u—_. .... .__._-. .‘__n—.n. m— .4...— The remaining case ofdim(Y) > dim(X) is of less interest, for it implies a degenerate relation“ ship between the components of Y. This degeneracy, due to a deterministic relationship between the many components of Y, would exhibit itself through the appearance of delta functions (unit impulses) to the resulting fy - 8.8 “oummnnvl "we cxetmse what hno EXEItCIsss mm a If X is measured in degtees centigrade and we wish to convert to Y measured in degrees Fahrenheit (1’2 32 +- (9/5)X), then determine fy in terms of fig b. IfX ~‘N(m 19), what is Y? \ 248 Chapters Functions of Random Variables ._......_ ._.. -u...“_..u.t.... .m_._m.,_._._. ..__..._..._...—...___-.___—._.— .. .._....._....__..uu.-—...... E82 a. If . r=1- -—X2, 13:05): iilU(x+1)U(1—x), then evaluate fy (y) (Hint: Start from the basic method.) b. Repeat for X ~IJ(-—1 1). E8..3 If X~ 3(4, 2) and Y: (X-— -,2)2 then evaluate and sketch the cdf P1207). EBA a. If X~ ~Par (a, 1:) and Y" -X.2, verify that Y is also Pareto Par 03, 1,") for some fl and t". b. Does this conclusion generalize for Y: X”? E85 a. L8! Y— '“" 3" Find Fy,fy in terms of Ex, fx. 1). Specialize this result for fy to the case of X~ ~'N. (Y is said to have the lognormal density, and this is used as a model for received power at the handset of a mobile cellular wireless system due to fading.) 138.6 a. If . F —1'ie—xU dY-—- 4‘ x00 — fir; (x), an — e , then evaluate the cdf Fr. . b, If we quantize X distributed as in (a) as - - 3 if'2<X Q: -_1' ifO.<_X5_2_,. ' ‘ mi ifXSO- then evaluate arid sketch the cdf FQ (q). E85l IfX~ ZJ( -- §," 2 ), then show that Y: a tan(X) has a Cauchy density fYOI)= 7r iaz + y5 1388 Time-sampled speech with amplitude X, having 15(06): ~-e ‘W, IS paSSed through a 2- bit quantizer qz if'x > r ' . '41 if'0<x ST- --~qz if :-I 5x 5 0’ —'qz' - if’x < "1'." with q(—::) =F-q(x)r- a. Determine the prof for the quantizer output Y: (10!). b Design the threshold '5 so that all four quantization levels occur with equal probability. use We are given thatX ~u(— ;,- g) and y: X2U{X). a. Determine P(Y= 0). b. Determine fire) for y 50.. 249 Exercises c. Determine FY (y) for y > 0.1 (1. Determine fy (y) ESJO If the voltage V~ N(O 0’2) 15 full-wave rectified to Y: IVI, determinefy.. E8.“ If the voltage V ~NI(m, 0.2) is half-wave rectified, Y: VU(V), determine fy ‘ 1518.12 If Y=X2U (X) X~ N (0 0'2), then determine .fi’O’) (Hint: Fy (0) ¢ 0.) ' E8.13 a. If Y: g(X), where ';a fix< a 300: x iffo 5a . I. a Iif'_x, > a then determine Piaf} 1n terms of FL. 13:. b. Specialize this result to the case where X is Laplacian: so that fir(x)— - —-e M. new n Y*.X+1 nxso fsxmr nX>o and fi;(x)-=; -e “,"I then evaluate fr for lyl < l. - E8.15 Measuring the power in decibels of. a ysignal X rs done through = 101°810CX2)« If X ~N(0, 4), determine the cdf FY (37). - 128.16 The thermal noise voltage in a resistor cf 1 Q at temperature T, when measured in a bandwidth W. has variance 02. We wish to represent the thermal noise power P: V_2 in decibels through the formula . P = 10103100?) Evaluate the density-I)";1 (x) E8.17_I A so called artificial neural network is composed of interconnected nonlinear nodes of the form ' In order to understand'the behavior of“ such a node, calculate fy, given that X ~ N011, a2)” E8.18 In a wireless communication system, the power P received by a mobile communication device at a given distance from a-basc station is a random variable that is modeled as log (5-) = X, X N N(0, 022); Pu ‘ where p”, is a given constant-knewn as the median power. ' a.‘ What is the density Ifiog(p)(z) of' log(P)? b What is the density I fit: (p) of the power P? ‘ a _ E8t19 In measuring speech signal power in db, we calculate Z = clog(A2), where A is the amplitude, log is the natural logarithm, and c = 10/ log(10).. , E820 E821 [£8.22 E823 E8..24 E825 E8..26 E837 E8.28 £8.29 E830 E831 Chapter 8 Functions of Random Variables a. What is the probability mode} for A? b. Evaluate the cdf 172(2). ' c. What is P(Z > 2c)? Show how to generate Y~13(3,1/2) from X ~ man. We ate given X~u(0, 1) and wish to simulate Y with P(Y =1/2-)= .3, P(Y = 2) = .25, P(Y 2- 3): .45.. Explain in detail how to do so. Given X ~LI(0, 1) find a function g, Y = g(X), to use so that 0 if"): <0 FY00 = y2 if-O-s y-< 1 1 if): 21 Show how to generate Y with edf‘Fyoz) = (l-—- e"”2)U (y) from X ~ MD, 1).. You are given X ~ “(0, 1) and desire a random vatiable Y with cdf 1 Pym— - {Mn How can you create Y? 'We are given a random variable X~ m0, 1) and wish to simulate a random vatiable Y with pdf fyO’)-—— - -y2U(1 —- iyl). Explain how to do so. - For a simulation study ofWWW file sizes, we need to créate Y with Pareto pdf Pay (a, 1) . Show how to do this, starting from X~u(01)- Given X ~ “(0, 1), show how to create Y with Fr 02) = (i..- if?) He). Determine the function 3, so that if X N LI(0,-1), then Y = g(X) hascdf . My) = (1. “—- e"?‘)U(y). - We are given X ~ Mo, 1). Explain how-to simulate Y with ' use) = évmva --y) + £602). _ (Hint: Sketch the cdf F12.) Explain in detail how to conveit X ~14“), 1) into Y with _ fro) =4y3U(y)U(1 1’). a. Given X ~11“), 1), construct Y with pdf' 1 n' I‘M)"- _ [vcosm if M. < 5 0 otherwise b. Construct z from X with piz = 1) = .6 and P(Z = 0) : Exercises - , \251 E832 3. Given X ~ M(0, 1), construct Y having density function My) = 29-23100). «. b. Given X as in (a), construct Z~ ~B(2 1/2). 138.33 If fx{x)_ — 2xU(x)U(1- -x), Y: g(X), then find g so that Y~ ~50). E834 a. If . fX1..X2(x1.-IZ) =’ e""."'2i‘2'U(x1). =X12 +112, 13 = 31g2 X2, then determine . fy1 ,yz. b.. Evaluate P (Y1 + Y2 2 0).. E835 a. Caiculatethy (x, y) for 1 . FLY (X. 3’) = Til—gag; e—J’ .. b. If W;X+Y., V szsz, - solve for X, Y as functions',of' W, V. c.. Calculate .fW v (w, v). d._ Provide an expression for fv(v). E836 A random power P: V1' is produced by voltage V and current I described by. fv 1 (v, y) - _ 6“" 'J'U(V)U(?)- a. Provide an expression for the edf Fp(z) by using our basic method. (A sketch will help.) b. Evaluate P(P < 0). c._ Prbvide an expression for the pdf' fp. £8.37 Provide an expression for the z'pdf fp (x) of the power P— 12R dissipated by a thermal noise current] of variance 0'2 passing through an independent randomly selected resis— tance R ~ “(r0 -- ‘1. to + 1) for re >_ 1.. (Hint: Calculate F}: from the basic method.) E838 You are given thxdxhle = e"'“2U(x2)U(x1)U(1 ”161). Y1= X1X2.- Y2 f—“ ---- a. Evaluatefybyz. b. Evaluate P(Y1Y2 > 1).. E839 An artificial neural network uses nodes of the'form _ _ ' r- ;_"z—e"z_ __ 2 Y —tanh(w X), tanh(z) — e3 +e'z — 1 1+e7-Z' for a given vector w Observe that tanh(z) is increasing in 2. Calculate fy, given that X~ MG), C.) (It is easier to first calculate fz, Z: w] X.) 252 1138.40 E8.41 E8 .42. 138.43 138.44 138.45 E8.46 I 1318.47 Chapters Functions of Random Variables An oscillator- X (I) = A cosmr + G), has a fixed amplitude A- and phase (-3: 0, but an unstable random frequency 52'” N000. 0-2) Determine 15((,)(x).. (Hint: Go back to basics and start with a sketch) The output Y of an AM radio envelope detector is fif +X22, where X1,X2 are i i d M0 02) (i e fx1.x2(x1.x2) 1%1(x1)firz(x2), X1~ ~N(0 02)) If Z ~ tan-1(X1/X2), then evaluate fl! 2 (Hint: Cartesian (X1,X2) to polar (Y, Z). ) Let X1,X2 be i i d. 8(a) (product factorization of the joint density as in the preceding problem. but with Xg~ E (01)), and Y1: X1 +X2, Yz— ._ X1 /Y1. Calculate the joint density 1‘le ya and verify that it is also of the product form I fyl (yflfy2 (312) We know that 1 .fiIJ.X2(x11 x2) = Ee-cxlz+x%), Y = 3X1 -+- XE. a“ Augment Y by a properly selected random variable Z so that you'can fairly easily determine fr,z(y. z)‘. b. Provide an expression for the pdf'Ify(y)., - a. If Y = Xle, then use augmentation to derive an expression for fir as an integral when we assume that fxl .Xz is known. ['1 Evaluateffiy) for O < y < 1 when X1 and X2 are both Ll(0 1). If Y: X12 + 2X2 with X1,X2, 1' 1‘. d N“), 1) (see the preceding problems for i. 1‘ ..d ), then evaluate fy. If X: A cos(wt + GD), where cu is a fixed constant, A and (9 are independent random vari- ables 2(meaning that fIA 9(a, 9): Ifi4(a)fg(6)) with A being Rayleigh distributed, fit (a) = —e ‘%U (a), and (-3 m U(- -11. 11), then evaluate the density of X0. (Hint: Augment this problem and use Iacobians) This problem arises when one attempts to understand narrowband Gaussian noise, a process commonly encountered 111 communications systems. The density would be the same for X, In communications, we encounter a signal S represented' 111 terms of amplitude A and phase (-9 though S- A cos(®). The amplitude and phase are such that 1.1.901 . 9) = f1 (aye (6) (independent) ' 1 . . 1511(0) = ae‘giz'ma), 109(3) 3 71 If M! S 11 .. 0 . otherwise Select an appropriate augmentation random variable Z and determine the joint pdf' Ifngz(s-,z), being careful to specify the ranges of validity. Exercises 138.48 a, The random power P supplied is the product of voltage V and cmrent-I (P m VI), with - .fV [(V,H)={ Select an augmentation tandom b‘ Detenninequa)” ' v-t-‘u if0$v51,05u51 0 A otherwise " variable Z and evaluate the joint pdffp_z(p, 2).. . 253 ...
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