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Unformatted text preview: .283  Exercises Theorem 9.18.1 A matrix A is a correlation or a covariance matrix if and only if it is
Hermitian and nonnegative deﬁnite.  Proof. We establish only the necessity of this candition. Say,
A = EXXT,
Then . ‘
sin xx'fa = maxim = Eelxxaixﬁ = 1322*, where Z = 21“)! is a scalar random variable. However, EZZ’“ = E [Z 2 2 0. Thus, A is nnd if it
is a correlation (or covariance) matrix. The Hermitian property follows from‘ A = EXXT => at: Etxx*)*_. — EXXT,
where we used the idempotent property OF)“: . i . E] An important Special case of a nonnegative deﬁnite matrix is a positive deﬁnite (pd) matrix
that satisﬁes the strict inequality a¢g=saiisa>o Lemma 9.18.2 A Hermitian matrix A is positive deﬁnite if and, only if all ofjits eigenvalues
are positive. In this case, A has a positive detetminant and therefore has an inverse.  . Proof citation. See Horn and Johnson [46], Theorem 7.2.1.. . , . . _ ' E] EXERCISES ‘E9.1' If
e(x>*=2xvcx)U(1—x). _ . then evaluate EX and VAR(X) _ 139.2 Iffx (x) .— ngOc  1), evaluate EX, VAR(X) 139.3 . If P(X= O)_ — .,2 P(X =1): 5, P(X= 2):. .3, determine EX, VAR(X). E94. IfP(X '=_—1)= .2 P(X= O)“ ._ .3 P(Xm 3)— ..— 5, evaluateEX, VAR(X) E95 Verify the table of expected values in Section 9.3.. E9.6 Verify the table .of' Variances in Section 9 10. ' ‘ ' ' E9.7 Evaluate EX 3X2, and VARCX) for X distributed as 6(a) and also as Par(oz z) ' Repeat for X~ POL) by ﬁrst evaluating EX and then EX (X » 1) E9.8 If P(a 5X 5 b) = 1, then what is the largest possible EX, and what is the disnibu
tion that achieves it? Also, what is the largestpossible .VAR(X) and the distribution
that achieves it? (Hint: Consider Y: X — (b +a)/2 the relation between VAR(Y)
and VAR(X), the maximum value of El’2 when iYI < (b— —a)/2, and the inequality
EY2 > VAR(Y)) . . , . . .. . 286 Chapter 9_Expectation and Moments E99 E910
E911 E912
E913
E914 29.15 E916 £9.17 E918 E919 E920
E921 E922 a. If T is a random temperature, will its mean ET be higher'when it is reported in centigtade rather than in Fahrenheit degrees? b. Will its variance be higher when T is reported in centigtade rather than in Fahrenheit degrees? If X ~ ma, b) and Y~ 30!, p), then evaluate EXk EY" for 1c 2: 0, 1,2, 3.
If X~ ~,LI(O l) and Y_ : ex , then evaluate EY ﬁrstly by use of property E8 of Section 9. 8 
and secondly by computing the density fy and using the basic deﬁnition of 131’ from Section 9 3.
Evaluate the mean and variance for X ~ ma, b). If If X~ ~,U(O l) and Y~ N(2, 3) then evaluate E(iX + Y). "mle" ifx<0 Ian): 1 . .
r—Ee‘” ifxao then evaluate VAROI). _
If the joint probability mass function Px.1r(0 0)= nlpx.1*(0 1)= 3 PIN/(1 0)— «,2 PXJ(11)= 4 evaluate the prof px (x) and E(XY). 
A component that fails without aging has a lifetime I“ such that its median lifetime {.6 is 1,000 hours. What' is its expected lifetime' in hours, and what rs the variance of that lifetime?
Find E'p for the power P dissipated in a 4" £2 resistox across which is a voltage V ~.,N(13). a. b.
a. b.
_c. a.
b. C A speech signal has Laplacianm distributed amplitudeX such that VAR(X)—— .— 2. What
is fit? If the Waiting time T to the next photon emission has ET: 3, what rs f7? If the mean square lifetime ELZ: 4 for a component that fails without aging, what
isP(L>3)? .   ' ‘ _ If V is a thermal noise voltage with W22: 2:,2 what' rs the pdi fv (v2)? If T is a waiting time between successive keystrokes such that ET2 : co, ET: 2
and ‘ro— = 1 is the largest value such that P(T > 1'9)— — 1, what is the pdf, f: (t)?
We observe thatX~ Par(4. 1). Evaluate EX, EXZ, VAR(X). We observe that Y~ ~,u(o 1). Evaluate EX EXZ, VAR(X) We are informed that COWX, Y ) ~— 1/8 Evaluate E(XY ) Show by evaluating the lefthand side that COV(aX +b. cY +d)=acCOV(X, Y). The noise variance for the thermal noise voltage V produced by a resistor of R ohms at
an absolute temperature T, when measured eve: a bandwidth W, is given in terms of the Boltzmann constant k by 02.. — _4kTRW. a. Evaluate the expected powet P dissipated' in R under these conditions when it is
shortcircuited. Exercises E9.23 E925 E9.26 b. 9‘?” 909‘?” sap99‘s b.. 287 If, under the given circumstances, we have two resistors of R1, R2 ohms each, and
they generate uncorrelated thermal noises V1, V2, then evaluate E(V1 + V2)? If S” is the amplitude of a speech signal with variance 4, then evaluate P(S > i).
If EX —~ 1, VAR(X) — 1, E'Y  1.,5 what can you conclude about the relative sizes of EX 2 EY 2" We observe X ~ N'(2, 1). Evaluate EX, VAR(X), EX2 We wish to infer to Y~ ~6(.l) Evaluate EY, 3Y2 ,VAR(Y). We are informed that COV(X , Y) = 1/2. What is the correlation E (XY )1? Which of' the two estimators of Y given by Y; = 0.5x and Y2 =X — 1 has the lower mean square error?
We observe that X~ Par (4, 1). Evaluate EX, EXZ, VAR(X).. We observe that Y~ ~,ZJ(0 1).. Evaluate EX, EXZ, VAR(X).. We are informed that COV(X, Y) =_ 1/8. Evaluate E (XY). Find the linear least mean square estimator Y : aX + b.. We measure X having EX = 1, EX2= 2 and wish to estimate Y having EY .— 1 EY2_ — 3 We know that E(XY) =0 If we choose an estimator f: aX + b,
evaluate a and b to achieve the minimum mean squared error E (Y Y )2
If, for the information given in (a), we decided to use a— — b— _— 1 as an estimator, evaluate its performance E (Y Y )2 E9. 27 (lEit Quantization) Let _ E9. 28 You are given that E929 q} if‘X > 1: X~£l,‘Y= X: _..
()  g( ) {go otherwrse The random variable Y is a 1bit quantized (AID converted) 'ver'sion ofX with quantize: a.
bi ,
c.. threshold ,1: and reconstructiorr levels 0 < qo < 41.. Find F,» and the mean and variance of Y.
Evaluate the mean square approximation error E (Y — X )2. We wish to design a 1bit quantizer, by the choice of 1:, qo,q1, to minimize the mean squared error. Show that, for given go, an, the best choice rs 90+91
. 2 . T= EY:3, EX: ,4 VAR(Y)=1, VAR(X)=2, COV(X, Y)'—05. a. If we decide to estimate Y by Y: 2X + b, choose I) to minimize the mean square
error. 
b.1f we decide to estimate Y by Y'— _ —X +1, what rs the mean square error of this
estimator?
c Using the information given at the outset, evaluate the correlation E (KY) and EX 2
We are given ' _ 3x2 if0<x<l,0<y<ll
fx.r(x.)’)— {0 otherwise 288 _ _ Chapter9 Expectation and Moments  Evaluate EX" and 151’".
Evaluate VAR(X), VARO’).
Evaluate E (X Y ). ' Evaluate COV(X, Y).
For given a, evaluate E(a(X— —EX) ~ (r en)? Find the MMSE afﬁne estimator Y (X) of Y based on X. If an oscillator
X0) = Acos(_toot + 9), (9 ~ LITrt, a"), A ~ S(a).' ‘ ﬁt. 96:13 9) =.ﬁt(a)fe(9). then evaluate EX (t) COV(X (t), X (s D. .VAR(X(t)).
13. Design the linear, least mean square predictor X (t) of X (I) based upon observing X(s) for some 1 < 2.
E931 We know that I ~ 8(1), and we wish to approximate to 1'2 by a linear function aT + b.
a. Find the least mean square coefﬁcients a“ ,b* by carrying out the minimization
calculation.
b. If a 1. 1/212: 0, then evaluate the mean square approximation error.
139.32 We wish to infer Z = Y 3 from observed X and know that the joint density ﬁg? (1:, y) is
equal to 1/2 over the rectangle 0 5 x S 2, 0 5 y 5 1 and is 0 elsewhere. Evaluate EZ, VAR(Z). Evaluate COV(X, Z) " ' Evaluate E[(Z— 452) — a(X— —EX)}2 as a function of :51. Find the linear minimum mean square estimator Z: cX + d of'Z. 1» rhea«99‘s 199.30 9957.” E933 _' x+_y if0<x<i, 0<y<1l
.fx.r(X.y)= {0 otherwise Evaiuate EX" and EY". Evaluate VAR(X), VAR(Y).. Evaluate E (XY ). Evaluate COV(X Y ).. For given a, evaluate E(a(X— EX)  (Y— —Er))2. Find the MMSE afﬁne estimator Y(X)= aX I b of Y based on X. E934 We observe that X: S + N with _ .
' s ~M(G,1),NN~N(0,0'2), cows N) =0.
a. Determine the linear, least mean square estimator S(X) for S. b. Determine the performance of S. E935 A voltage V~ ~80) is measured by an instrument having response K” V +N that
contains an uncorrelated normally distributed error N of zero mean and variance 02 52999575” Exercises E936 E937 E938 E939 139.140 ..........._.——._—__..._...—._......... ...._...._..____..—__.———.._l__._.._.._._..i We attempt to infer V from Y(X_) = aX + b., What arethe choices of 41,!) that yield a minimum mean square estimate? .
The two dimensional random vector X has covariance matrix 2
a a 0'
C3 = 1 p l i
_ : 90'1“: 0’2
The random vector Y: ,1UX where U' is the linear transformation of rotation through an
angle 9, or
U ._. c059 — sin9
_. sine cost) " Evaluate the covariande matrix (Cy.
In a certain linear system the inputs X are related to the outputs Y through the formula Y: AX, where
' 1" 1
A: (M1 1)” . 2 1 .
RX ‘ (1 3)?
ﬁnd the correlation matrix Ry for the outputs b. Can you evaluate the covariance matrix for Y? If  _ . ' 7 _.
 :1" 5 2
X~N '2 , ' 2 .10 ,
_ _o —3 —6 13 evaluate H2 and COV(X1,X3)
In a given circuit, the currents I are related to the voltages V through the formula we: awo : 1‘ ' _ 4 2
Ev: (2) cv =C0V(V,V) — (2 5)“.
a. Evaluate Eli. ‘ i b.‘ Evaluate cova,r)='cc,._ .' In a twomesh resistive network'with two noisy resistors, the relationship between the
two mesh currents I and the two equivalent noise voltages V is given by the formula (.2 2122 2)'v la.If It is known that 290 chapter 9 Expectation and Moments We know that Evaluate f1(i1.i2)..
E9.4l a If Z: (1;) NJV(m,iC), m: ((1)) ,
. 1 .3
«2:512Hyman—u):r = 1 2 ,
 — 2
2 evaluate EX , EY, COV(X , Y), VAR(X); and VARG’ ).
1).. Evaluate the mean square error perfOrmance of 71 Y = 5(X — 1) ' as an estimator of Y ..
E942 For 1' = 1. 2, we observe that X. = S +Ng, where S N N(m, 1_),N,v ~N'(0, of), and
COV(S, M) = COV(N1, N2) = O.‘ . . a. Evaluate COV(S,X,—).‘
b.. Evaluate the covariance matrix C3 = COV(X, 152).. E9.43 The random veetor .. X, N  _ o _ '2 —1
xo ()( 3)] a. Evaluate EX, VAR(X1), VAR(X2), COV(X1, X2)”
‘0. Let X2011) = aX1+ b be a linear estimator of X2 based on Xli. Select a and b to minimize E (Jig — x2)?
c Evaluate the performance of this Optimum linear estimator: d" If _
I 2 Y = AX, A = 3 —1 ,
—2 2 evaluate EY and COV(Y, 1’). 1319.44 Show, by applying the deﬁnition of positive deﬁniteness in terms ofquadt'atic forms, that
if we select an arbitrary n x n. realvalued matrix A and any 6 > 0, then H=A7A+ei is an n x :1 positive deﬁnite matrix. Exercises .  291 1519.45 Given X with 3 x 3 covatiance matrix II, show how to synthesize Y having covariance mauix
2 0 O
CY : 0 1 0 ..
0 0 3 E9.46_ If you start with X having covariance matrix 2 0
C3 = (o 1)’
then show how to synthesize Y with covariance matrix 2 —l
(. 3). E947 Starting from the Maﬂab command randn(3 . 1.) ,_generate a Gaussian random vector
Y with EY = (0 —1’ 1)’ and covariance matrix 1.6340 0.3459 0.6776
Cy: 0.3459 4.0900 —'1.5354 ., 0.6776 «1.5864 1.5411 ,: x. ...
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