Chapter 07 Problems - Exercises 223 Restofing the terms in...

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Unformatted text preview: ' Exercises ' . , 223 Restofing the terms in fathat depended upon x1, 1.1.1, xn_1, we find that .. n+0ln+1wln ' . F 1111-] .n 2 a ,. f‘(x1,.‘ .‘ ..,x,._1) -_-. ..__(_);_t.-;l_°‘i)_._._.(1_.zxk 1—1ch at 1“ k . k=1 _ Thas,lf'(xj, t, xn_1) is precisely of the form of a DitichIet 230)) when we identify (We 5 n — 1) fit: ==¢¥k7 fin =59»: +€¥n+h as was to be shown. EXERCISES EM Let each F} be a univatiate cdi and define the bivatiate function FOE, 3’) = 01-551(I)F2(J’) + 01-5F3(X)F_4(Y)1- a. Evaluate limbs“, 3.1.00 F (x 'y). b. Is liqum F (x y) a nnivariate cdi‘? c. Evaluate 62P(x, y) ,f(x,y J’)=-—axay— n d. Isf a bivariate pdf‘? en. Is F a bivaxiate cdf? _ 137.2 Let each F.- be a univatiate cdf and define the tiivariate function F (x y Z) = wFi(x)F2(J’)F3(Z) + fiF4(x)F3(Y)Fis(z)» a What are necessary constraints on a and fit for F to be a cdf? b Assuming that these constraints am satisfied, evaluate the hivatiate cdf F (x 2:) that is determined by F (x, y, 2) E73 If the bivariate cdf F}; m y) = o 3¢(x1)U(y + 1) + ..7U(x)(1 -- e‘33’)U(y) ((I> is the N(0, 1) cdf), then evaluate the joint ptobability that both— -2 < X < —1 and Y >—— ——-..05 E7..4 The Jomt pdf finrtxJ) = e“U(J_c)U(v)U(1-y)n Evaluate the probabilities 01A. A“, and B for? A = {(X,-Y.) :X' 2 Y} and B = {(X, Y) :X .-= Y}.. 224 E75 E726 137.7 E73 E73 Chapter 7 Multivariate Distribution Functions and Densities .._.... _._._.....-..... ...._m-...___ ..-__-..._. .u. A joint or bivariate pdf‘ 91 9957:» b.‘ C 3::2 ifO<x<1,O<Iy<1I fit “x y) _. {0 otherwise Evaluate the pdffy ‘ Evaluate the edf F y Evaluate P(X_ > Y ) Is this joint density of the product or innovation type? If the density fo y(x y)... — e 2” U (y)U(x)U (2 x), then evaluate the density fig and the cdf Fx Is this bivariate density of the product or innovation type? Evaluate the probability that X exceeds Y, P(X > Y). In a particle accelerator, a pointlike particle strikes a circular target of radius 1 The impact point is equally likely to fall anywhere within the target area ‘ a. bi. o, d.‘ What is the density fir 1-0:, )2) when X, Y are Cartesian coordinates for the plane of the target with the origin at the center of the target? Is this joint density of the product or innovation type? . What is the probability that the particle will strike within a distance r of the center of the target? Evaluate the cdf Fx (x) (for the .x-coordinate for x < 0).. There are two components C1, C2, having random lifetimes L1, 14, respectively, that have the density aux. y) = «newts-Wows), where (1(2) is the. unit step function and or, B are positive. a. b.‘ 'c“ d.‘ Is this joint density of the product or innovation type? Evaluate the cdf‘ Fir-£12" EvflnmethecdffkiofLL Evaluate the probabilities of the following three events: - L1 = :1; L1 > t1; both components fail prior to time t.‘ nmunfiymn=Wn FxriLy) =' (1 ----- e"“‘)(’1'--e'“"'>U(x)U<y), then evaluate the joint density fig'y, the marginal cdt no) =P({(X. Y) :X 5x». and its density fin Exercises £27.10 E'Lll E112 E7.l3 E7..14 a.‘ If 0 ifx<0ory<0 - 1 ifxe- 1,y>1 Fx,r(x,y)= xy if'05x51,05y.<_1_. x if0'5x51,y>1' y if'05y51,x>i then evaluate Fx‘. b.. Evaluate the pdffinyu We are given that ' __1_ i ' _ 2'_ 2 firm?) -— 2t_i(x)<i(y) i- 2”UH Ix y )3. a. Verify thatfi” is a pdf. b., Evaluate and sketchjk. c. Evaluate and sketch the df F}; a. Evaluate P(X=’- + Y2< 4).. If the joint pdf 'x+ if0<x<l,0< <1. mey>={0 y .3’ . otherwise then evaluate P(Y_ > X -+ 5)- 225 Ihe joint pdf', for waiting times T1, T2 to the first and second particle emissions, is given by fn r2031. Ire) = aze‘“"2U('t2 -" Ii}U(t1) = { a Is this joint density of the innovations type? by. Evaluate the pdf'f-j-I (11). e. Evaluate the pdf ‘ {1202). (firm if :2 a n a 0 0 otherwise " In many circumstances, we measure a signal X that is a sum S +N of a quantity S that we are interested in and a measurement noise N I When S and N are unlinked, it can be shown (see Section 13.5) that in this case the joint density fans (I. S) = 3%‘(8')fN(x "- S)“ . a. Is fx s of the product or innovation type? b If N ~ N(0, 02) (a frequent assumption about measurement noise) and S NN (1115,02), evaluatefij. c.‘ Isfins ofthe bivatiate normal type? 225 ' Chapter 7 Multivariate Distribution Functions and Densities E7.15 If the joint cdf' . 71 . rmcx, y, 2> = 1—;‘33';m';:- then evaluate P (X :3. 1).. (This does not require a lot of calculation.) E116 a. Show that, for any 'n 2 1 and positive a1,.....,,an and posifive 51’ ....., 5”, the function - . . 0061, .. .. .. ix”) = [1+ 2 new 1"" - . H is ajoint or multivariate cdfi. (Hint: Show that the corresponding density is nonneg-- ative, and use the properties of G to show that this density integrates to 1..) b. Is G of the product or innovation type? ' E117 Let firm, x2) be a Dirichlet pdf with (21 = a2. a. Is fa a symmetric function in that, for any x1 and .x2,fi,(x1,x2) =fa(Xg,x1)? b. If 0:; = a2 = 2‘ andag = 3, whatis the pdf for X1? E7.18 Given that X ~ flax]. a2) and Y ~ [3(053, :24), what are the constiaints on the {0:5} so that X and Y are bivariate Dirichlet? 1437.19 A message source has an alphabet of the three symbols a,b. c, and it generates a message of length 100. The symbols are chosen with respective probabilities pa, 125,, pc, and successive symbols are independent of each other or unlinked with each other. The symbol probabilities are unknown to us, but we know that they themselves were chosen according to a Dirichlet, with parameters are a 1, ab = 2, (2., == 3 for the respective prob- abilities. _ a. What is the probability that the message will contain 30 appearances of the symbol a and 25 appearances of the symbol 1:)? b. What is the probability that the same source will generate a sequence having 50 appearances of a? 137.20 A trivariate normal random vector X has mean vector 0 _ 6 7 2 m = —1 and covariance matrix C = ‘7 10 3 .. 2 2 3 4 ' a. Write out the pdf"fx(x;,x2, 263).. b.. Write out the pdfjjgqugx], x3). c.. Write out the pdffx2 (x2). (1.. Which of X2 or X3 has the greater probability of being positive? E7221 Let g1, 32, and 33 be univariate pdfs, and define ISLYZUJ’: z) = armed)! —z2)ga(x - w)“ a. Verify thatfxyz is a trivariate pdf and determine its type. b.. Determinefyz. c. Provide an expression ferfx. —-.._.m___.._.———u«-~_._ ._..__-___-- Exercises -. 227 137.22 a‘. If a system with lifetime L operates three subsystems in parallel mode, and the subsystem lifetimes L1, Lg, 14 are described by ‘ 3 Fz,,14.zg<x1.xz.x3) = 1‘10 Weave». ‘ i=1 ‘ then evaluate the overall system lifetime edf‘ FL(x) and pdf‘f; (x). b, Repeat this analysis for serial mode. e. Repeat this analysis for spare pans mode. [£7.23 The bivaxiate Cauchy for a > 0 is given by the bivaxiate pdf a _. . . 15am») = Ear-(“2 +x‘f‘ +_y2) i, a“ Verify that this yields the univariate Cauchy fx(X) = ——--—.‘ (I b. Hence, vefify that fly is indeed a bivariate pdff. ...
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