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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. Characterize-the 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 discrete-time 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
_ ---u2-u22+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) — 3-3—
1 1:22 "
a Evaluate £12..
bi. Evaluate'gbﬂu).
cw. Are V1 and Vgiindependent?
E1126 In a linear, discrete-time system the air-dimensional 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 function-qty;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 d-dimensional 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 motion-detecting, narrow-beam infrared (IR) detector looking at low-level 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 that-the 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 y--rays 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 char-acreristic function on for the random numberA of ant-particles 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 time-average 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 bit-lengths {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,X1r-M}, 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 GF-Gx (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