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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 ofj-its 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 iY-I < (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). "ml-e" 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 Laplacian-m 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 left-hand 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
short-circuited. Exercises E9.23 E925 E9.26 b. 9‘?” 909‘?” sap-99‘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 (l-Eit 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 1-bit 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 1-bit 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 ~ LIT-rt, 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 9-957.” 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 5299-9575” Exercises E936 E937 E938 E939 139.140 ..........._.——._—__..._...—._...-..-.... ...._...._..____..—-__.———.._l__._.._.._._..i We attempt to infer V from Y(X_) = aX + b., What are-the 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 two-mesh 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 21-22 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
x-o ()( 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. real-valued 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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