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Unformatted text preview: Exercises“ '   1 5 . 247 'I‘lteisecond case is that 1 2; 3’1 ”:3 #2,? " " 1" 3 13301;) 2“ The denizity f“ is G for 5 set) 311' ’for m. _... b. uu._.._.‘u—_. .... .__._. .‘__n—.n. m— .4...— The remaining case ofdim(Y) > dim(X) is of less interest, for it implies a degenerate relation“
ship between the components of Y. This degeneracy, due to a deterministic relationship between
the many components of Y, would exhibit itself through the appearance of delta functions (unit
impulses) to the resulting fy  8.8 “oummnnvl "we cxetmse what hno EXEItCIsss mm a If X is measured in degtees centigrade and we wish to convert to Y measured in
degrees Fahrenheit (1’2 32 + (9/5)X), then determine fy in terms of ﬁg
b. IfX ~‘N(m 19), what is Y? \ 248 Chapters Functions of Random Variables ._......_ ._.. u...“_..u.t.... .m_._m.,_._._. ..__..._..._...—...___.___—._.— .. .._....._....__..uu.—...... E82 a. If .
r=1 —X2, 13:05): iilU(x+1)U(1—x),
then evaluate fy (y) (Hint: Start from the basic method.) b. Repeat for X ~IJ(—1 1).
E8..3 If X~ 3(4, 2) and Y: (X— ,2)2 then evaluate and sketch the cdf P1207). EBA a. If X~ ~Par (a, 1:) and Y" X.2, verify that Y is also Pareto Par 03, 1,") for some ﬂ and t".
b. Does this conclusion generalize for Y: X”? E85 a. L8! Y— '“" 3" Find Fy,fy in terms of Ex, fx. 1). Specialize this result for fy to the case of X~ ~'N. (Y is said to have the lognormal
density, and this is used as a model for received power at the handset of a mobile
cellular wireless system due to fading.) 138.6 a. If .
F —1'ie—xU dY— 4‘
x00 — ﬁr; (x), an — e , then evaluate the cdf Fr. .
b, If we quantize X distributed as in (a) as   3 if'2<X
Q: _1' ifO.<_X5_2_,.
' ‘ mi ifXSO then evaluate arid sketch the cdf FQ (q).
E85l IfX~ ZJ(  §," 2 ), then show that Y: a tan(X) has a Cauchy density fYOI)= 7r iaz + y5
1388 Timesampled speech with amplitude X, having 15(06): ~e ‘W, IS paSSed through a 2 bit
quantizer
qz if'x > r ' . '41 if'0<x ST
~qz if :I 5x 5 0’
—'qz'  if’x < "1'." with q(—::) =Fq(x)r a. Determine the prof for the quantizer output Y: (10!).
b Design the threshold '5 so that all four quantization levels occur with equal probability.
use We are given thatX ~u(— ;, g) and y: X2U{X). a. Determine P(Y= 0).
b. Determine ﬁre) for y 50.. 249 Exercises c. Determine FY (y) for y > 0.1
(1. Determine fy (y) ESJO If the voltage V~ N(O 0’2) 15 fullwave rectiﬁed to Y: IVI, determinefy..
E8.“ If the voltage V ~NI(m, 0.2) is halfwave rectiﬁed, Y: VU(V), determine fy ‘
1518.12 If Y=X2U (X) X~ N (0 0'2), then determine .ﬁ’O’) (Hint: Fy (0) ¢ 0.) '
E8.13 a. If Y: g(X), where
';a ﬁx< a
300: x iffo 5a .
I. a Iif'_x, > a then determine Piaf} 1n terms of FL. 13:.
b. Specialize this result to the case where X is Laplacian: so that ﬁr(x)—  —e M. new n
Y*.X+1 nxso
fsxmr nX>o and ﬁ;(x)=; e “,"I then evaluate fr for lyl < l. 
E8.15 Measuring the power in decibels of. a ysignal X rs done through = 101°810CX2)« If X ~N(0, 4), determine the cdf FY (37). 
128.16 The thermal noise voltage in a resistor cf 1 Q at temperature T, when measured in a bandwidth W. has variance 02. We wish to represent the thermal noise power P: V_2
in decibels through the formula .
P = 10103100?) Evaluate the densityI)";1 (x)
E8.17_I A so called artiﬁcial neural network is composed of interconnected nonlinear nodes of the form ' In order to understand'the behavior of“ such a node, calculate fy, given that X ~ N011, a2)”
E8.18 In a wireless communication system, the power P received by a mobile communication
device at a given distance from abasc station is a random variable that is modeled as log (5) = X, X N N(0, 022);
Pu ‘ where p”, is a given constantknewn as the median power. '
a.‘ What is the density Iﬁog(p)(z) of' log(P)?
b What is the density I ﬁt: (p) of the power P? ‘ a _
E8t19 In measuring speech signal power in db, we calculate Z = clog(A2), where A is the
amplitude, log is the natural logarithm, and c = 10/ log(10).. , E820
E821 [£8.22 E823
E8..24 E825
E8..26 E837
E8.28 £8.29 E830 E831 Chapter 8 Functions of Random Variables a. What is the probability mode} for A?
b. Evaluate the cdf 172(2). '
c. What is P(Z > 2c)? Show how to generate Y~13(3,1/2) from X ~ man.
We ate given X~u(0, 1) and wish to simulate Y with P(Y =1/2)= .3, P(Y = 2) = .25, P(Y 2 3): .45.. Explain in detail how to do so.
Given X ~LI(0, 1) ﬁnd a function g, Y = g(X), to use so that 0 if"): <0
FY00 = y2 ifOs y< 1
1 if): 21 Show how to generate Y with edf‘Fyoz) = (l— e"”2)U (y) from X ~ MD, 1)..
You are given X ~ “(0, 1) and desire a random vatiable Y with cdf 1
Pym—  {Mn How can you create Y? 'We are given a random variable X~ m0, 1) and wish to simulate a random vatiable Y with pdf fyO’)——  y2U(1 — iyl). Explain how to do so. 
For a simulation study ofWWW ﬁle sizes, we need to créate Y with Pareto pdf Pay (a, 1) . Show how to do this, starting from X~u(01)
Given X ~ “(0, 1), show how to create Y with Fr 02) = (i.. if?) He). Determine the function 3, so that if X N LI(0,1), then Y = g(X) hascdf .
My) = (1. “— e"?‘)U(y). 
We are given X ~ Mo, 1). Explain howto simulate Y with ' use) = évmva y) + £602). _ (Hint: Sketch the cdf F12.) Explain in detail how to conveit X ~14“), 1) into Y with
_ fro) =4y3U(y)U(1 1’).
a. Given X ~11“), 1), construct Y with pdf' 1 n'
I‘M)" _ [vcosm if M. < 5
0 otherwise b. Construct z from X with piz = 1) = .6 and P(Z = 0) : Exercises  , \251 E832 3. Given X ~ M(0, 1), construct Y having density function My) = 2923100). «. b. Given X as in (a), construct Z~ ~B(2 1/2).
138.33 If fx{x)_ — 2xU(x)U(1 x), Y: g(X), then ﬁnd g so that Y~ ~50).
E834 a. If . fX1..X2(x1.IZ) =’ e""."'2i‘2'U(x1).
=X12 +112, 13 = 31g2 X2, then determine . fy1 ,yz.
b.. Evaluate P (Y1 + Y2 2 0)..
E835 a. Caiculatethy (x, y) for 1 . FLY (X. 3’) = Til—gag; e—J’ .. b. If
W;X+Y., V szsz,  solve for X, Y as functions',of' W, V.
c.. Calculate .fW v (w, v).
d._ Provide an expression for fv(v).
E836 A random power P: V1' is produced by voltage V and current I described by. fv 1 (v, y)  _ 6“" 'J'U(V)U(?) a. Provide an expression for the edf Fp(z) by using our basic method. (A sketch will help.)
b. Evaluate P(P < 0).
c._ Prbvide an expression for the pdf' fp. £8.37 Provide an expression for the z'pdf fp (x) of the power P— 12R dissipated by a thermal
noise current] of variance 0'2 passing through an independent randomly selected resis—
tance R ~ “(r0  ‘1. to + 1) for re >_ 1.. (Hint: Calculate F}: from the basic method.)
E838 You are given thxdxhle = e"'“2U(x2)U(x1)U(1 ”161). Y1= X1X2. Y2 f—“  a. Evaluatefybyz.
b. Evaluate P(Y1Y2 > 1)..
E839 An artificial neural network uses nodes of the'form _
_ ' r ;_"z—e"z_ __ 2
Y —tanh(w X), tanh(z) — e3 +e'z — 1 1+e7Z' for a given vector w Observe that tanh(z) is increasing in 2.
Calculate fy, given that X~ MG), C.) (It is easier to ﬁrst calculate fz, Z: w] X.) 252 1138.40 E8.41 E8 .42. 138.43 138.44 138.45 E8.46 I 1318.47 Chapters Functions of Random Variables An oscillator
X (I) = A cosmr + G), has a ﬁxed amplitude A and phase (3: 0, but an unstable random frequency 52'” N000. 02)
Determine 15((,)(x).. (Hint: Go back to basics and start with a sketch) The output Y of an AM radio envelope detector is ﬁf +X22, where X1,X2 are i i d M0 02) (i e fx1.x2(x1.x2) 1%1(x1)ﬁrz(x2), X1~ ~N(0 02)) If Z ~
tan1(X1/X2), then evaluate ﬂ! 2 (Hint: Cartesian (X1,X2) to polar (Y, Z). ) Let X1,X2 be i i d. 8(a) (product factorization of the joint density as in the preceding problem. but with Xg~ E (01)), and Y1: X1 +X2, Yz— ._ X1 /Y1. Calculate the joint density 1‘le ya and verify that it is also of the product form I fyl (yﬂfy2 (312) We know that 1
.ﬁIJ.X2(x11 x2) = Eecxlz+x%), Y = 3X1 + XE. a“ Augment Y by a properly selected random variable Z so that you'can fairly easily determine fr,z(y. z)‘.
b. Provide an expression for the pdf'Ify(y).,  a. If Y = Xle, then use augmentation to derive an expression for ﬁr as an integral
when we assume that fxl .Xz is known.
['1 Evaluatefﬁy) for O < y < 1 when X1 and X2 are both Ll(0 1).
If Y: X12 + 2X2 with X1,X2, 1' 1‘. d N“), 1) (see the preceding problems for i. 1‘ ..d ), then
evaluate fy.
If X: A cos(wt + GD), where cu is a ﬁxed constant, A and (9 are independent random vari
ables 2(meaning that fIA 9(a, 9): Iﬁ4(a)fg(6)) with A being Rayleigh distributed, fit (a) = —e ‘%U (a), and (3 m U( 11. 11), then evaluate the density of X0. (Hint: Augment this problem and use Iacobians) This problem arises when one attempts to understand narrowband Gaussian noise,
a process commonly encountered 111 communications systems. The density would be the
same for X,
In communications, we encounter a signal S represented' 111 terms of amplitude A and phase (9 though S A cos(®). The amplitude and phase are such that 1.1.901 . 9) = f1 (aye (6) (independent) ' 1 . . 1511(0) = ae‘giz'ma), 109(3) 3 71 If M! S 11 ..
0 . otherwise Select an appropriate augmentation random variable Z and determine the joint pdf'
Ifngz(s,z), being careful to specify the ranges of validity. Exercises 138.48 a, The random power P supplied is the product of voltage V and cmrentI (P m VI), with  .fV [(V,H)={ Select an augmentation tandom
b‘ Detenninequa)” ' vt‘u if0$v51,05u51
0 A otherwise " variable Z and evaluate the joint pdffp_z(p, 2).. . 253 ...
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 Spring '05
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