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Unformatted text preview:  Exercises m. 433 B—measuiable and that agrees almost surely with Y5 (P (Z = Y3) = 1) will also satisfy the deﬁni
tion of being a conditional expectation Hence, theie aremany versions of conditional expectation,
but each pair agrees almost surely. Obseive how the awkward case 'of' conditioning on an event,
say, Be 6 B, that is null (Pale) = 0) is handled. with apparent ease in this deﬁnition of conditional
expectationwthere is no division by zero” It is clear from the deﬁnition that any choice of Y3(co)
for a) 6 Bo is acceptable. Conditional expectation E (Y 13) can be deﬁned arbitrarily on a null set in 8“ However, we still need constructive means to identify Y3? EXERCISES _ E15..1 at If ﬁg y (x. y) = e“""” U (x)U (y), then evaluate E (Y X)‘, .
b.. If many) = %1e‘x"U(x)U(y), then evaluate E(Y1_X)..
E152 If (Vi)EX,2 < co and (Vi aé j)E‘(X, 1X3) = 0, then evaluate E311 and E(X1X2)t
E153 We observe voltages V1 and Y2 having a joint density that is uniform over the unit disk; ' Evaluate E (V2] V1)“ _ _ I  .
1315.4 a. If we know that E(V2V1) = V13 and that V1 '9 L“ _—_1; 1), then evaluate the cone. lation between V1 and V2. by. Are V1 and V2 independent?
E155 a. IfX  m0, 1) and E(Z IX) = X2, then evaluate EZ.‘
b.. For the speciﬁcation of part (a), evaluate the 'couelation E‘(XZ)..
E156 a“ If, in a system with output Y and input)! ~ 8(a), we know that E(Y IX) 2: X2, evaluate EY t. 
b.‘ Evaluate the input—output correlation E(YX)U .
E153 It is known that X ~ ‘N'(O, 1) and 'E(Y_lX) = X4” Giving reasons for your answers, are
X and Y independent or uncoirelated? 
E153 It is knOWn that; for all x, E_(l’ {X _=x) > x‘. .
a. If' EX is ﬁnite, what can you conclude about E_Y and EX?
b.. If EX is inﬁnite, what can you conclude about EY‘?‘
E153 A receiver whose antenna is pointed at the ‘sky yields output voltage measurements V1
' and V2, made at different times, that are jointly normally distn'buted with mean zero,
equal valiances of 1/2, and a covariance of 1/4. :1. Evaluate the conditional density ‘ fvzlvl y.
b.. Evaluate the minimum mean square estimator V2(V1) of V2” 1215.10 If _ _ _ _ _ __
me,» = (x +y)U(x)U(y)Ut1x)U(1y). . then evaluate the minimum mean square estimator f7 (X ) of Y .. . 
1315.11 We obsetve the product)! = SN , where the signal S ~uc1, 2) and the independent
noise N has the Pareto model '  7 he) = ﬁzzle — n,  434 Chapter 15 Conditional Expectation a: Determine ‘ fxls (xii). (Hint: Consider the cdeF'xls ‘) . b.‘ Determinefstﬂslx)‘. Be explicit about the range of' s and the cases J: < 2 and on. What is the minimum mean square estimator .§' (X) of the signal given the obser
vation; for observations x Z 2?  r d.‘ What is the performance of this estimator? ‘ e.‘ Determine the asymptotic behavior (large sample size n) of E (S; — S)? E1512 at. If Y = ex +N and XJLN, with X «Mon, a2), N ~ Mo, 1), provide an ex.
pression for the MMSE estimator )2 (Y) of X given Y M .
b.‘ What is E}? (Y )? (This can be done with very little calculation.)
E1513 A thermal noise voltage V appears across an independent random resistor R N JV(R0, '1),
R0 >> 1, and it induces a noise current 1..  ' at. If EV? = 2, what is fy? b.‘ Evaluate the joint pdffygﬁv, r)‘. . .
Evaluate the average power E(V2/R) dissipated in R4.
Evaluatefm (v, i).for the current I through Re. Are V and I independent? ' ' '
What is the MMSE I‘m!) that estimates R from V? E1514 If ’9’ 2bit?!“ ' fscx) = 2sU(s)U(1— s),N.._LLS,N ~ £_(1),X 4—. s +N, Y = 52, then ﬁnd the minimum mean square estimator 17' (X) of Y .i
b. Evaluate the performance of this estimator” .' . "
£315.15 A system transforms its input X ~ 8(a) into an output Y through ' . ' 2  _ We wish to br'edict theoutput Y from the input X i _ an Evaluate the minimum meansquar'e error predictor 1711’)“
.b.‘ .. Evaluate the mean square error performance of this estimator. 1315.16 at. If'
1 (y~xzxz)2,  ﬁssurin Ix: . x2) =. 752 with {X.} it'd N‘(0; a2), thendeterrnine the least mean square estimator )9 (X1, X2)
of Y ., .  ‘ . :  bi, Evaluate the performance of 1‘". (Hint: Consider subconditioning.) E1517 a. A noisy measurement Y =X +N of X ~ 190.1) arises from additive indepen
 » dent noiseN' ~ Poo); Evaluate PU = 35).: ' ’ ' ' [1. Describe what we know about X after observing Y i. c.‘ Evaluate the least meansquare estimator J? (Y) of X given Y i. d.‘ Provide an expression for the performance of'1i(Y).. /\ Exercises 1315.18 13115.19 1315.20 E1521 E1522 435 . at It XJLY, X ~N(1,2.), r ~.N‘(—1, 1) and z =XY, then
VAR(Z)H 
bi. Evaluate COV(X,Z),  ‘
c1. Evaluate the linear least mean square estimator XL(Z) of X H d.‘ Evaluate the least mean square estimator 20:) 012 given X. (Caution: Note the
changes from part (c)..) ‘  ‘. _ ' _  .
rr Y = mile, where N.lL{X,—},' {X.} are stag; ~ mo, 1), and N ~ 190.), then
evaluate E (YIN) and HY” '
a. Using the matrix operation of trace, express the error criterion E(X —— iFM
(X  ii) in terms of the positive deﬁnite weighting matrix M and the unweighted
error covariance matrix ]P = E (X  It)? (X —_ 12).,
bh Simplify the expression when M is a diagonal matrix with diagonal elements
{di}«  . ‘_ .
Consider the emitter location problem of Exarhple 155, only now we have just a single
sensor X1 at (0, 0)‘. Provide an expression forthe MMSE'estimator if of the emitter ' location Y.
A dynamical system with state vector Xk at time 1;; is speciﬁed by ¢=randn{3.3), B=randn(3,3), evaluate EZ ' and with initial condition Xo satisfying ' ' nxo=o, C0V(X0,Xo)=iPo=]I
and input process {Uk} satisfying _ ' o ﬂj¢k EU}: : 0. COVtUj.Ur) =.= {E if]. =k I The observation process 215k _>__1, is speciﬁed by 0211 if“ j = k
. _ 0 ' if j :1;
Once ‘seiect'ed; <1), B, IHI shouldbe treated as nonrandom, The initial condition X0, mea 1111': ranan(2,3), EVk =0. COV(Vr.Vri) ={ ' surernent'noises,'_and input noises are also uncorrelated with each other: ' a What is the leastmean square estimator of _
bi. Using.__Matlab or any other computing environment, for k = 1 : 100, iteratively evaluate _ .._ Pr = coves “taxi — it)“ c.,~ Using your computing environment, fer k = I : 100; iteratively evaluate 12,;
(Zr, ~ ’11)"  , . '_ , . d. For each of the three compenents of the state vector, plot the true state and its ' Kalman estimate, bath as a function of the time index k” ' ...
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 Spring '05
 HAAS
 Conditional Probability, Probability theory, ex, Estimation theory, Minimum mean square error

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