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**Unformatted text preview: **Exercises biannual leads: to :a‘ ‘just-iﬁeatio ' of the bhiomial; {Seefﬁecti‘ ' Application is given, to t and to expectation of-éprodutzts in Section 17 T old 3 provide If
that is computaﬁonalljiiisef EXERCISES 1214.1 If 9 = {1, 2, 3, 4, 5, 6],A =' {1, 2, 4}, and B = {2, 3, 5}, then construct a-pmprc) such thatAlLB nontrivially. :
1214.2 If S2 = {0, 1, t. L. .t , 9} and pk = 01, what are the independence relations between the three events _
A = {0, l, 2, 3, 4}, B = {2, 3, 4, 5,6,7}, C = {0,1, 2, 6, 7, 8}? E143 The event F of_ system failure occurs if either A; or A2 occurs, but A3 does not occur;
- . If it, A; and P041) = .4, P(A2) = as, and P(A3) = .1, then evaluate P(F)., _
E1444 Consider Bernoulli trials with arbitraty P(X.- = l) = 19.. Group the odd and even trials
{(ngﬂ, X210}, Let N 2 0 be the index-of'the ﬁrst k_ for which Km“ 9E X2”, _ a. Detemne P(X2N & 1), _ ., 7 _ _ .
‘ b. can this process be used to create the equivalent ofa truly fair coin?
E145 We are given that .lLT-A; with PM; =-- _ _
I a, If B is the event that at least one _of' {A5} oncurs, then evaluate P(B).
__ - ' b, If C is the event that either-At or: A2, but not A4; occurs, then evaluate P(C).,
. 1314.6 Let X,- be 1 there is an error in the ith bit and bit otherwise.._Assume that X = {X,-}
are i it!“ (independent and identically distributed) with P(X.- = 1) = p. Consider the
events . _ . IA={-XI:X1-=X3 == 1}, B ={X:X4=Q}_,
' Cetx=xzao,'Xs=1t-. an Evaluate the probability P, that all three events oocm, - b. Evaluate the probability E; that none of them occur;
on. Evaluate the probability 1”c that at least one of them occurs.
d.. Evaluate the probability Pd that A and B, but notC, occutz El4.7 Recall the network displayed in Figure 2.5.. Let 0],, k :- 1 : 14, be the events that the
link between nodes k and k + 1 is intact. Assume that the. events {Ck} are if with
probability p. Recall that all of the links are unidirectional from the lower numbered node to the higher: - - a. What is the probability P1“; of a connected path from node 1 to node 6?
b. What is the probability 95:15 that there is- no path from node 6 to node 15? ‘ 1214.10 1314.11 E14.12 E14.13 1314.14 E14.15 Chapter 14 Independence Urn U,- has r,- :i red balls and g,- = 10—1" gieen balls. In an unlinked experiment 80,
we select U,» with probability (131)055 for i = 1, .t .. t. , 6.. We then choose a ball at random from the selected um. What is theptpbability that the ball is red?
A given source produces an inﬁniter long sequence of messages {Mil that are chosen ' in an and , fashion with P(M.- =' m) = pm forthe choice of” the speciﬁc message m on the ith trial Consider two distinct p'osSible messages mo and m1“ a“ What is the probability that message m will appear before message m on the
nth trial? by, Vetify that the probability that eventually message mo will appear before message
"31 is PMs/(Pam +Pm.)« ' _ ‘_._1 i£05x.51,yao
fxw(x'y)u{2x if'Oﬁxsl, y<0’ then are X , Y independent? _ . - -_
'a The input X to a system 8 with output Y is such that
' 13(02): e_"U(x) and POT my) for}: e 1.2, W...
What can you say about 8? .t _ _ _ -
b.‘ For the setup of (a), if you also know that X .lL 1’, what can you say about 8?
A system is subjected to Lind“ shocks X1, t. v. ,Xn with X,- ~ Ll(0, S)., Ihe system fails
if a shock exceeds 3». Find the probability of“ system failure," ‘
a. System A is composed of a *‘sexies’.’ connection of components 0;, .t .t .. , Cn- and
fails toopexate if any cemponent fails“ The timeto failure of C,- is If the {7;}
A are Lind“ with density 8(a), then evaluate the density for the System lifetime T.‘
- by. System B iscomposed of a “parallel” connection of the same components as
System A, and B failsto operate only if" all of its Compenents-fails. Evaluate the
density for the lifetime T of System 3.. ' ’ ' ' '
A system is composed of three Components 0;, C2, and C3. having respective lifetimes
L3. 14, and L3, with corresponding probability laws L.- N 8mg). If the individual compo-
nents fail independently of each other-and if the system fails if either of the components
fail, then what is the density f; for the overall system lifetime L? -
A system 8 having lifetime L iscornposed of three subsystems 781, 82, and 83 having
independent lifetimes-L1, I4, and Lgthat are identically distributed with continuous cdf'
F .t 8 fails if either=61 fails or if both 82 and 83 fail. ' _ -
a. If A]? denotes the event that Lgrsx and B denotes the event that L 5 x, then
express B in terms of {Aj 1.. 7
Evaluate the odf F; (x) for 6".
Evaluate the pdfﬁ... - - ' -
Is P(L1 < 14 < D3,) = P(£3 < L1 < Lg)? (This iseasy, with a little thought.)
Evaluate the probability P; that 5' fails due to the failure of 81 -(i..e.‘, P1 = P(L1<min(L;_,Lg))n . 99953" Exercises £14.16 ' E1447 1314.18
E1439 j E1420
1214.21 E1422 E1433 E1424 E14..25 E1426- 1314.27 E1428 41.1. A radar range detector Works by considering the outputs {X.-} from each of r- “range
bins,” with a target being identiﬁed as at range r; if It} islarge. When no target is
present, the outputs are i..i .d. 8(a). A detection occurs if any of these r outputs exceeds a threshold r. What is the probability of false alarm? - t
We observe X;, in l :n, i..i..d . 21(0, 6).. One way to infer 6 is to form the estimator- 2
n 1 Evaluate the pdffz,1 (z).
If‘X ~ 3(1), Y ~ “(0, 1) and they are independent, then evaluate P(X > Y),
A system composed of :1 components having individual lifetimes L; at are i .i..d_.. 8(a), fails if any component fails. ' a. What is the probability of the systinn lifetime r exceeding $1?
b. Evaluate the density function fr of the system lifetime I IfX, Y are i..i.ld.. 8(1), then evaluate P(Y > 2X). I A random experiment 6' consists of choosing an integer from the set {1, 2, .. .. .. , N}, with
each integer having equal probability of being chosen. Independent rcpetitions {5;} of
this experiment are conducted until the ﬁrst time '1‘ that the number chosen equals one
chosen in a previous experiment. Find P(T' = t). ' ' ' Calculate the probability that no two people in a group the size of your class section
have the same birthday. (Neglect leap year and seasonal considerations.) Approximate
log(l x) by —-x to obtain a simpliﬁed answer. See if any two people present share a
common birthday. ' . ' Ir .tLj‘Xi, X; ~ .N'Om. of), then evaluate the joint density ‘ fx, $213 and theconditional
dﬁnsitYﬁnIJa-t . ' 7 I . .. . .. Xi, .i..i..a'.. fx(x) = é—e‘i"! are inputs to a discrete-time integrator rtht. Yt=Xt+n_r(k>1>.. ' a- Partially describe the system response by ﬁndingjizl 3333...:
33.. Find fxaiyz. ' X;,X2,'X3, i..i..d.. 8(1), and
Y1=X1X2X3, Y2 =13th Y3 ¥——X1--' Findtf-Yl 12.1301..th In an i'..i..d..'model for errors made in transmission of text symbols,- T is the waiting time (number of symbols) to the ﬁrst occurrence of an error. If the probability of' an
individual error is .1, what is the pmf' p; (k)? _' ' - A photodetector receives a photon count Z arising from the sum oftwo independent
light sources having photon counts X; N 170,-),1' =1, 2. Evaluate pz, pzlxl , pxllz. A binary symmetric channel has crossover (error) probability p.._and Operates inde-
pendently on its binary input symbols {X.-} to produce its output symbols {Yr} (e..g.,
(X1, Y1)_i_L(X2, Y2». The input message source can emit only one of the three binary
sequences m1 = 00, m; :01, and m3 = 11, with respective probabilities .5, .3, and. .2. E1429 E1430 Chapter 14 Independence a. Find the probability that the output Y1 Y2 = 10.. ' .
b. Determine the conditional probability of each of the three possible messages
m1, m2. and m, given that we receive 10. ' '
c.. Given any received sequence ylyg, the minimum error probability decision as
' to the message that was sent is to choose the one with the largest posterior
probability P(M = Inlele =y1y2). If p = .2. and 10 is received, then what
should our decision be as to the transmitted message? An analog channel adds independent noise N N .N(0,2) to its input X to preduce
its output Y. The “input is distributed with cdf' Exec) = ..2U(x + 1)+ .617 (x) +
.2U(x — 1).. _ - ' ' - r. ' a. Determine the channel characterization fylx. b. Determine the channel output response ‘ fx ..
c. What is the most probable input value if the output is 1, and how probable is it? A system S having lifetime L is composed of two diﬁ’er'ent components having individual
lifetimes L1 ~ 5(1) and L2 ~ 8(3). These components of S fail independently of each , other. . - E1431 E1432 ' ' £14.33 E1434 1:14.35 E1436
E1437 a. If S fails if either component fails,then evaluate ‘ 11. b. If“ S fails only if both _components fail. then evaluate ‘ fl.
IinL‘Y, IX '~ 8(1), Y ~ 11(0, l), and Z = XV+ Y, then deterrninefzu).
a. If ' - ' " ' - X(t) ammo: +9), an o. 0 ~ U(—r_r, x), A ~ 8(a), then evaluate EX (t), COV(X(t), X (s)), and V§R(X(t)).
b. Design the linear, least mean square predictor X (I) of'X (t) baSed upon observa-
tion'ofX(.s) for'some i <1. '- ' - . " ' _ ' ’
A system 5 having lifetime L is composed of four subsystems 81, . . .', S4 with lifetimes
L17, .. . .. L4, respectively, that are i .1‘ ..d. 5(a). 8 fails if both 81 and 82 fail or if both 83 and S4 fail. . ' ' . a. Letting A = (L 5 r) and A.-
b.. Determine the cdf F; (1)..
c. Determine EL. We ObserveX = S +N, I
5 .1L N, N ~N(0,0'2), P(S = —1) ems = 1) .—._ ms =' 0) = .5. 3.. Evaluate ﬁns ., b.. Evaluate‘ﬁg. '
c. If‘X : 1,-then what do we know about S ? = (L,- 5 1:), express A in terms ofAh .. . .. ,A4. ' if at: you know is that P(X 5 so! = 2) < P(X 5 2n’ = 3), what can you conclude about the independence of‘X , Y? (Explain) 7 -
If"ﬁg,y(x,y) =‘1/n on the disk of unit radius andﬁ;n'y(x, y) = 0 otherwise, is X .1}. Y? .IfX, Y,Z are i-..i..d.. and,- individually, .N(0, 1/2), what is jig/,2? Exercises E1438 1214.39
1314.40 E14.41 13114.42 15114.43 E14.44 E14..45 E14.46 413 A sequence of random variables {Xk} is recursively generated through X020’X1-3lexz =0X1+N2. Xr =an--1_+Nr. where {N1c }' are inr‘ “d” with common pdf‘fN” a. ISNkJLqu?’
. b. Is NleXk? _
e, Provide an expression forkalxhl fork a 2.‘ a. We know that Lian,- P011) g= ..4,P(A2) = ‘4, and P(A3’) = “5.. Evaluate P(B)
. ' for B the event that all three of these events do not oecur;
bi. Repeat (a) for P(C), Where C is the event that either A1 or A3 occurs.
The random variables X1, 4. .r .‘ ,Xn are initdn m0, 1)), '
a. If Z = mare-5,, Xi, provide an expression for E2 .
b“ If Y = minisﬂ Xi, provide an expression for EY‘. The random variables X}, .‘ ,X,, are it d.. 8(a)).
at. If' Z = mans" Xi, provide an expression for EZ.‘
b.. If Y = minign Xi, provide an expression for EY ‘
We observe X = (X1, 4.1. UK”) Lind“ PO.) and wish to estimate it from X.
a. Determine the MLE estimator RX)“
bi. If you are now informed that A ~ 8 (1), provide an expression for the mean square
error made by MK)
We observe X 2 (X1. 4. r. .. ,Xn) rid). B(no, p) and wish to estimate p from X for
given no. a. Determine the MLE estimator ;3(X).,
b.) If you are now informed that 19 ~ “(0, 1), provide an expression for the mean square error made by ﬁ(X).,
We observe x = or], .r ,Xn) rid. 9(a) and wish to estimate a ﬂour x.
at. Determine the MLE estimator 3(X)..
b.. If you are now informed that 5 ~ 11(0, 1), provide an expression for the mean
square error made by 50‘). We observe X = (X1, .. .r .‘ ,Xn) rid. 8(a) and wish to estimate or from X).
I a. Determine the MLE estimator &(X)., _
b. If you are now informed that or ~ 8(1), provide an expression for the mean
square error made by 6200. _ '
We observe X: (X1, .r.‘.‘,X,,) Lia". JV (111,02) and wish to estimate m from X for given 02‘, at. Determine the MLE estimator rit(X)..
b. If you are now informed that are ~ Nit), 1), provide an expression for the mean square error made by 54X)“ Chapter 14 Independence [$14.47 We obseriIe X: (X1,v.v..‘,Xn) 12m". -N(0,v) and wish. to estimate the vatiance v
from X.‘ a” Determine the MLE estimator i300.
b.‘ If you axe now infozmed that v '~ 5 (1), provide an expression for the mean square error made by 1700.. _ _
£114.48 We observe Y : X +N as the output. of a measurement system 8 having input
X ~ PG) and additive independent N ~ 8(a). _ -
. 3. Evaluate P(X 5 1’)” (This is easy, given a little thought.)
b. Specify what we know about X, given that we have observed Y.. A
ci, Provide an expression for the enfor probability MAP estimator X (Y)
of X given Y that is valid for all Y‘. ' I
d. Evaluate X (Y) fot‘ 'Y = 1.5., ...

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- Spring '05
- HAAS
- Conditional Probability, Probability, Probability theory, Mean squared error