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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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