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Unformatted text preview: Eercises ______l__ ______ _________v __ ______ _____I mm
EXERCISES.
E111 If'P(X : l) = ] —— P(X = 0) =12, evaluate ¢x(u). _
E112 a.. Compute the characteristic function (bx (u) for X ~ u(——a, a).
b. Compute the characteristic function for the Laplacian, X ~ £(or).
E17.3 a. Compute the characteristic function for X ~ B(n_, b. Compute the characteristic function for X I~ 13(1). ‘
E114 If X isan integerhvalued random variable with P(X = k) .= pg and characteristic func
tion ox, then show that (bx is periodic by evaluating nix (u)  ¢x(u + 29:). (Carry out
_ any simpliﬁcations.) .
E115 If ax (u) =' 1 /(1 + :12), evaluate VAR(X).
E1745 If ‘ I . i 1 . . 2
_ . . Q¢X(u)=(1+u2) °
then evaluate VAR(X)..  7
E117 Determine EX and VAR(X) from the characteristic function axe) = (I Etc—2.. E173 a.. Evaluate EX, EXZ, and VAR(X) for X f.“ B(n, p) from its characteristic function.
b. Repeat part (a) for X 5 POL);  ' . '_ . .
E173 Evaluate EX, EXZ, VAR(X), and EX3 from the characteristic function for X ~ .N5(m, 02).. E1110 a. If a random variable X has the characteristic function
., I . p 1 _ . 2 i . _
axon = '—£~~'—9‘)—”'— 'for' 0 < o: <1, 1 + a2  2a cos(u)_’ then evaluate EX and VAR(X.)» ' 
b. If {X1,X2} are i Li .d.. as X of part (a), with o: = .5, then evaluate the joint char
acteristic function ¢x(u) =; ¢Xlgx2(u1, H2)..'  . 
E17.ll a. Professor Phynne claims that l M“) = In? is a characteristic function. Evaluate VAR(X) from 411;.
b. Is Professor Phynne correct that d; is a characteristic function? (You must give
_ precise reasons for your answer.) _ 14117.12 a. Show that (bx (u) = e"“‘ cannot be a characteristic function by evaluat
. ins 5X2”  . . — . .
h. Which of the properties of characteristic functions stated in Section 17.3 do not
hold? ' '  a '
E1113 A Brownian rnotion particle with mass in moving with a velocity'V has kinetic energy K»1 3 V2
.. i=1 480 Chapter 17 Characteristic Functions where the components {Vi} of V are i ..i..d.. N(0, (:2). Evaluate the characteristic function
(bx in terms of 45,12. (Note that the density for 2K/m0' is chi squared with three degrees of freedom, or x?)
[317.14 If
axe) '= e‘2“1""i‘"22+"1"2~, then evaluate EXlX2, and 4521' (u).  
E1115 A photodetector' is exposed to'the light from three unrelated sources N1, N2, and N3 of respective intensities II , 12, and 13..
3.. Making appropriate assumptions, provide the characteristic function on for the
total number N a N1 + N2 + N3 of incident photons in time T.
it. What is the probability that zero photons will be observed? '
c. What is the most probable number of photons to be observed? E1116 3. The random variables {X1,...., X5} a1’ez'.i..d. Assuming knowledge of (ﬁg, deter:
' mine the characteristic function dis” (u) for the weighted sum . 1 ,, _
urn—ﬁght,  1:. Determine the characteristic function for 5,, f asn. ' _ ,
E1117 A system 8 having input X_ adds independent noise N {v 5(1) to X to produce its
' output Y. _' ‘ . ' _ _ ‘ ' _ . '
a. Characterizethe system by calculating? ﬂax”
b. If the characteristic function ofthe output is
‘ ' '  ‘l ,. i . _ 2
62"“. 1+ uz. ¢r Cu)" = I then what is the input density ﬁp? __
E17018 A discretetime integrate: accepts inputs {Xi} and yields outputs
n .
“HY” = Z ' V II ' j=ﬂ—IL . . , I
'a. If{X,} are independent with ¢Xi (u) = aiZ/(af + :12) for a. > c (the Laplacian),
then evaluate 41y". _ ' .   ‘
I ' b.‘ Evaluate EYn and VAR(Y,,) by using (ﬁn.
' E1119 In a certain system S, the _i_np_ut_X and output Y have a joint characteristic function . ¢X 'Y (1" V) = e’10+loewe—A1+llel(u+v) I.  ' :1. Describe the output 1’ by providing“ its characteristic function
b. Evaluate EY. c. Evaluate the conelation.E'(XY)..I
d. Determine the characteristic function ¢z (u) for Z = Y X. t 481 Exercises E1120 The two voltage sources in a circuit have amplitudes V1 and V2, with means ofl and a
covariance matrix I
' 2 1
cv _ (l 3) a.
There are two circuit currents I; and 12, Ielated to V through I: GY, where c r.; (j ' a. Calculate El and the covariance matrix CL
b.. If V1 and V2 are jointly normally distributed, then evaluate the characteristic function 451‘
' E1121 3.. If' Y1, Y2 is the response of a'linear system given by
__ Y1 _ 1 2 Z; N H _~
I Y _ (Y2) _. (3 I) 21.11.22, 21 N(1,2), 22 mo, 5),
then evaluate EY.‘ ‘ ‘ . _. . . '
bl. Evaluate the joint charactetistic function W01): ¢y1,y2(u1, u;) for Y in the
preceding equation. E1122 In a given circuit V = 2!, we know that
_ u2u22+u u . ___ I "2 .
¢1(u)— e l 12, Zn. 3 a. Evaluate C0V(Il,12)‘
b‘. Evaluate q>v(u)t. E1723 In a given circuit, the mesh cutrents I are related to the source voltages V through a conductance matrix _
' 2 —1
G: v—l 3 , I;GV.,_
0 —1 a. If the source voltages ate Lind. 'with common characteristic function the (u) =
(FM, then evaluate ¢v(u ..
bl. Evaluate (by  ' _ E1124 In a given circuit, the mesh currents I are related to the source voltage V ~ N(m, 0'2)
through _ I _ . . 
_ 1+ [I] I Determine the charactetistic function m. I  _'_ .
E1125 In a given circuit, the node voltages V are related to the source cutrents I_. through the . impedance matrix . ‘
x v =' m. z = (“.2 Ti) ~ 482 __ Chapter 17 Characteristic Functions We are informed that "2
ecu) — 33—
1 1:22 "
a Evaluate £12..
bi. Evaluate'gbﬂu).
cw. Are V1 and Vgiindependent?
E1126 In a linear, discretetime system the airdimensional vector of state variables Xk at time
k is related to the vector of State variables X“; at the next time k + 1 through X0 = 30. Xr41=Axk + Zt+r for k 2 0. where A is a known nonrandom square matrix and the random vectors {Zk} are i..i..d.r with common characteristic function '
T _ ¢z'(u) = e'" “a.
Evaluate the characteristic functionqty;1 of X1” 1317.27 We have 2 2
emu1 nu; 1 + (ting—)2" .‘ 951312 (In. “2) = Evaluate ¢YI and ¢y2n .
Is Y1J1.Y2? Evaluate EYl.‘ Evaluate VAR(Y1)., If 9999'? I z __ Z] __ Y1 ‘— Y2 '
_ 74 “_ Y1 + 2Y2 . ’
evaluate 4321 2201,122)” I ' . E1128 2.. If the characteristic function dam) for a ddimensional random vector X is
' ‘given by '  ' ' ' ¢x(u) = exp (i Z kuk ‘ 21mg?) .
' 1 I I k=1 kml
then evaluate the characteristic function for the second component X; of X. b. For the setup of (a), evaluate EX: and VAR(X2). E1129 If‘X, Y are i ..i .d . Cauchy, verify that 2X = X +X has the same characteristic function
as Z =X + Y. (This is a curious instancein which the sum of two iridi. random
variables is identically distributed to a sum in'which the two variables are completely ' dependent.) I ' ' ' ' E1139 A motiondetecting, narrowbeam infrared (IR) detector looking at lowlevel backgron optical radiation produces an output Y that is the difference X1  X2 of two in? “d” photon sources. What is the characteristic function 45;; (u)? Essa... ._...______..._._,__.___n__.___.____...___..___;._n.n_..:4§.§
E1131 Verify thatthe Laplacian (3(a) is the correct probability law for the difference D a
11 — 12 in intensity levels between adjacent pixels by postulating that the intensity levels _ are Lid 8(a)“ : _
E1132 In an Aloha protocol, two callers whoso packets have collided are assigned t' id” retrans
mission times R1 and R2 that are distributed as u“), r). Retransmission is successful if R1—R2]> 0. at Calculate the characteristic function 42th
b“ Calculate the characteristic function 41:2 for Z = R1 — R2” 317.33 If {is} are Lind, Cauchy with characteristic function eriul and 3,, = (1 /n) )jjg‘ X;, then
evaluate the characteristic function 923,,” (Note from this thaL‘in the absence of a ﬁnite
mean, the average of even a very large number of initd. random'variables need not
stabilize or converge to a constant!) ' E1134 {X} are i..r".d.. with‘P(X == 1) mp = 1 — P(X = 0)“ a. . Consider the average  . 1 n I .
rt = ' Exit
n 13:1
and evaluate its characteristic ﬁJnCIianbAn (u) _ r ‘ _
b. Determine the limiting valueof m" (u). as It grows, and conclude that the disuiw
bution of the average converges to a particular constant
c.. ConSider the normalized sum '_ .
.. I n
Ilsa: “Pl, [=1 and evaluate its characteristic function (#5,, (u). '_
(1; Determine the limiting value of ¢Snr(u) as n grows, and identify the corresponding
distribution“ ' _ ‘ '
E1735 In a compound Poisson model for Unix netstat, wehave a number“ C’ of client processes,
each of which generates numbers N; ofpackets that are 1' Mind” 'POL.) and are independent
of Ci. The total number of packets NI: Mif C_ > 0; and'N = 0 if C :0. The characteristic function cm. (u) = tar—WM“). a. If C = c > 0, determine 961.; and repeat for C = 0‘
b., If C" ~ POI), then evaluate P(N = n)" . E17.36 The number N of yrays incident on a particular_‘materia1 in time T is random and described by a 9(ﬁ) distribution. The ith incident y—ray produces a'random number X, of a—particles, where X. is described by a characteristic function ¢x(u). Assuming independent effects from different y_—rays and making reasonable assumptions, evaluate the characreristic function on for the random numberA of antparticles emitted in time 1'“ ' E1137 The random number X. of electrons emitted from a particular material due to the ith
' ' incident photon is described by _ ' ' ' ' ' P(X, = O) = 1/3, P(X, =1) 7 2/3, 484 . Chapter 17 Characteristic Functions and the responses to different photons are independent. The number N of photons inci
dent in time T is random and described by the Poisson POJ') distribution 3. Evaluate the characteristic ﬁmction of the number 5 of emitted electrons in
time ’12 ._' ' _
b. Evaluate the expected value of the timeaverage current 1i
=— 'x
Ti=1l flouting in time T‘. E17238 In 'a given period T, a random number N of symbols are transmitted by a terminal.
The'probability of exactly n Symbols being transmitted is (%_)_"+1.. Errors are made
independently from symbol to symbol with probability "01; and the errors are made
independently of the number of symbols transmitted in T i 3“ Evaluate the characteristic function 465 (u) for the total number S of symbols in
error that have been transmitted in T. (It may help to let X; be 1 if there 'is an
error in the ith symbol'and be'O if otherwise.) ' b.. Evaluate E‘S E1139 A random number N of'packets of varying bitlengths {L,} are received in time T in a
certain digital communication network, where  ' '
1521‘ I "d elem”
N ~ ‘ ( >, in at. ‘ ¢1.<u)— ;; eel—1'" 
Evaluate the characteristic function for the total number B of bits received in time T.
E1140 For the design of a cache size, we need to know the amount of storage that might
be required to cover a period of T seconds. Assume that the number N: of ﬁle sizes
‘ requested is Poisson with an average of it ﬁles per second._Assunre further that ﬁles
sizes S, in bytes are find” with S ~ 903) (the ZetaIZipf“ would be more realistic, but
the characteristic function is awkward) and independent of the number of requests. Let
Bz' denote the total number of bytesreceived in time ' 'I ' '
I a. Determine the'expected number EBj of bytes arriving to be stored in time T..
b. Determine the characteristic function for By t. " E17.41 We have Y = XXI, .il.{N,X1rM}, N ~ mom ~ m1)“ a. Evaluate EY.
b., Evaluate qby‘.
_ c. Is Y Binomial; Poisson, or Geometric?
E1142 In a certain situation, the random opportunities for performing tasks are such that the
number N of such tasks that can be performed in an allotted time T is described by
N N 12100 (a uniform distribution over 0, 1, .. .i .. , 99)“ The ith task has a payoff X, with
P(X, = —1) :1: = 1 — P(X.— m 1)” N and X1,X2, .,.. .i are mutually independent. Exercises 485 :1” Evaluate ¢y (u) for the cumulative payoff N
Y =in.‘ i=1
b.‘ Evaluate VAR(Y ) from (35y.
E1143 If P(X .= 1) = 1  P(X = 0) = p, evalualethe generating function Gx(s )H
E1144 Using the table of characteristic functipns, determine the GFGx (s) for the following
and check your results by evaluating 0x0):
an X ~ 1201,); '
b‘ X ~ ramp);
c X ~ 'P(A).. ...
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 Spring '05
 HAAS
 Probability theory, Characteristic function

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