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**Unformatted text preview: **389 Exercises inference or-dwisinnémaiting memes
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conditionai probability at the measurement proeei EXERCISES E1.3..1 In a certain communications system, the received signal X is distributed 8 (1) if message
m is sent and is distributed 8 (2) it message mg is sent. Message m1 is sent 40% of the
time and m2 is sent otherwise. 7
a. Determine the receive: r310!) that has the minimum probability of error in deci-
ding onfthe transmitted message. '
b Evaluate the error probability for ﬁt. - -
E132 In a given radar detection problem, if the target is absent, then we observe X e: N(0, 1)
whereas if a target 'is present, “re-observe X N NO, 1).. Professor Phynne has solved
this problem for the decision rule H (X) having minimum'error' probability and found that
A b o l
H“) ___ a sent rf‘x <"1 H
- ' present if x 2 1
a. When is this the correct decision rule? A
b. Provide an expression for the error probability performance of this rule H when it is Correct. 1 _ ‘ - I
E133 a IfS is such thatP(Y :le =2) is TOO.) in k with A =1+x and P(X x 0) =
- .P(X = 1-) = 0.5, then describe Y1. . ' '
b.: Solve for the MAP rule X (Y). 313.4 A system .3 transforms X into Y such that I ,_1- 2- 2
fX.Y(xa)’)== £8 1&4")- a. Determine the MAP rule for inferring X from Y..
b, What is the error probabiiity P(X(Y) #X)?
1213.5 Iffmoa ix) = Ix + I]e""+””U(y) and X ~ 13(2, §), determine the MAP rule 20’) for . t Y :1. . , . - .- - . - . . -
E136 Consider theso-called Z—channel system, used to model binary optical communications,
in which the channel input X and output Y both take their values in {0, 1}, with the 390 I Cheater 13 _ MAP. MLE, and Neyman—Pearson Rules channel speciﬁed by _
_ 1 -_ if): = y n 0'
Prlxotlx) = 1--.D if'x =y =1 _
p ‘I if: = Ly m 0
Note that a 0 is always-received as a 0, but a 1 may be received either as a 0 or a It. -
The input process is such that P(X = 0) = no, P(X = 1) n my. ‘ a. Describe the channel output, ‘ b.. Evaluate the probability that the input is 0, given that the output is 0.. . c, Professor Phynne has proposal the decision rule X *(Y) = Y to recover the input
from the observed output, Evaluate the error probability performance P: of this rule
d. If Y = 0, then for what values of no and p is the r'ninir'mim error probability rule
X (0) = 0? ' ' ' 7 E13.7 A system 8 transforms X into Y such that ' 29“”? if 0 5 y 5 x
_ ﬁt'ﬂx’y) _ {0' I if otherwise ”
a, Describe 8, being careful to specify correct ranges of variables-x and-y,
bi. Given that we observe Y A: y,- what is known about X? -
- c. Determine the MAP rule X for inferring X from Y.
d. What is the error‘probabilﬂy‘ P(X (Y) ac X)? E133 In a certain communication system, the received signal X is described by a density
function e‘xU (x) if an is sent and by density 2e‘2" U(x) if' message m; is sent. a. Design alreceiver so as to maximize the probability that we decide that m; was
sent given that it was, subject to the restriction that the probability is no more
than or of deciding that ml was sent when it was net. _ K b. If message m is sent 70% of the time and message m; is sent the remaining
30% of the time, then. what is the probability of correctly deciding m1 when we
use the preceding design? ' ' ' 1213.9 If a system defect is present, then a monitor-will yield a voltage X having a density
3x217 (x)U(1 m x)” If there is no defect, then the monitor- voltage is described by the
density U(0,1). ,. ' - t ' a. Determine the Neyman-nPearson ROC for the problem of detecting the presence
or absence of a defect. ' ' ' ' b, ' If we desire a false-alarm probability of .2, then what is the corresponding detec»
tion probability? - ' 3 ' ' . . E13010 With a pseudonoise signal radar, if there is no reﬂected signal from a target, then we ' receive Y ‘~,N'(0, 1).. If a’t’arget. is present and reﬂects the signal, then we receive
Y~N(0,2)._ ‘ ' ' -- ' Exercises r 391 a.‘ Determine the Neyman--Pearson ROC for the problem of detecting the presence
or absence of the target by specifying Pp (a) and Pm (05)“ b. If you now learn that the prior probability of a target being present is ‘3, then
evaluate the error probability when you use a Neyman—PeaISOn decision rule for
a false---alarm probability of "1.. E13.11 A radar sends out a pseudorandom pulse and then waits for a given interval of time for
a reﬂected attenuated pulse. Ifa certain target is absent, then the radar receiver output
is described by X ~11“), 1).. If the target is present, then ﬁdx) = 4x3U (x)U (1 — x). a. Determine the optimum receiver to use for a given false-alarm probability.
b. Provide expressions for the false-alarm and detection probabilities”
c. Calculate and sketch the receiver operating characteristic (ROC).._ 1313.12 We observe a voltage V that arises from noisy reception of a, signal S drawn from
' {50, 51} V has the conditional density ' fvrs(vinl = c -+ 1)e"_"‘+‘1""U(v)'sntv)“ . a. Identify the set {v : decide S :51} for Neyman—Pear'son hypothesis testing“ by. Select the threshold 1: so that the probability or deciding s; when so was sent
(false alarm or Type 1 error) is at. - -- c.‘ For this '5, what is the corresponding probability p of correctly deciding s1
(detection probability or power of the test)? . _ . ' d.‘ If we have the additional information that P(S = so) = p0, then what is the
probability of a wrong decision? 7 ' ,~ - e. Evaluate P(S =st n v)“ El3.13 A binary-valued signal S e {—1,_1} is measured inthe presence‘of additive independent noise N ~ N(0, 1)‘. _ _ _ _ at Determine a decision rule based on the measurement X that incorrectly decides
that S = ml with a probability of "1 and maximizes theprobability of Correctly
deciding that S = —1r - - b. Evaluate the probabilities of correctly deciding that S = ---1 and of correctly
deciding'thatS = 1‘. ' - . ' . . . . I . E13.14 A binary-valued signals 6 {—1, i} is measured in the presence of additive independent
noise N, ‘ fit-«(11) = %e""'.‘ _ a. Determine a decision rule based on the measurement X that incorrectly decides
that S = 1 with a probability of Eye"? and maximizes the probability of correctly
decidingthatS=lm ' ' ' ' ." bl Evaluate the probabilities of correctly deciding that S i '—1 and of correctly
deciding that S = 1 for a decision rule 1 ifirx >0 ‘19): L1 ﬁx 5 o " ‘ ‘5 392 E1115 E13.16 . E1317 E13.18 E13.19_ ' E1320 Chapter 13 MAP. NILE, and Neyman-Pearson Rulg It may help to note that
2a if'x > a
2x u if Ix! 5 a. ..
«242 if x < —a We observe X '9 ’P(A) and wish to estimate it from X. a. Can you determine the MAP estimator ion? b.. Determine the MLE estimator ion.
c.. If'you are now informed that it ~ 8(1), provide an expression for the mean square I.lx-+'al-lx—ai= error made by 51(X). - ' ‘
We observe)! = S JEN arising from a signal S ~ 1300) and additive, independent noise N ~ Pm), with My: girl. -- . a. Specify pxlsocls).
b.. Provide an expression for p51); (s Ix). .
c.. Determine the MAP rule .§'(2) for estimating S from X when X = 2.. (Hint: Just
try the different possible values of S.) _ y .
d. Evaluate the error probability Pg insurred by using X itself to estimate S,
We observe X ~ B(n,'p) and Wish to estimate p from X for given n. a. Can you determine the MAP estimator 1300? b.. Determine the MLE estimatorﬁ(X)._.U . . _- .
c. If'_ you are-now informed that p ~ 24(0, 1), provide an expression for the mean square error made by 15(X).
We observe X ~ 9(5) and wish to estimate ,6 from X. a. Can you‘deterrnine the MAP estimator 30!)?
b.. Determine the MLE estimator 1300.,
c.. .If you are. nowinformed that [3 .~ mo, 1); provide an expression for the mean square error made by 5(X). -
We observe X r21302:) and wish to estimate or from X. a. Can you determine the MAP estimator MK)?
_ b.. Determine the-NILE estimatorﬂX). _ __
c. If you are now informed that a ~ 8(1), provide an expression for the mean square error made by MK)- ' _
We observe‘X-«i N(m, 0'2) and wish to estimate m from X for a given 02.
a.. Can you determine-the MAP eStir'nator' 751(X)? ' b. Determine the MLE estimator 51(X ).. _
' e. If you are now informed that :12 ~ .N(0, 1), provide an expression for the mean square error made by r3200. Exercises I 393 ..._..__——..._4.,v_....._...__—.._...—_._—.-_.._ ...... .—.._.-__..._...— E1321 We observe X ~ .N'(O, v) and wish to estimate the variance v from X .. a. Can you determine the MAP estimator 9(X)?
1).. Determine the MLE estimator 1700..
c. Ifyou are now informed that v ~ 8(1), provide an expression for the mean square error made by 17(X )..
E1322 We measure S and observe X . It is knoWn that 1 ' '
more) = Evade—‘1. no) = e‘svts). a. Evaluate the MAP rule .5 (X )..
b. Evaluate the NILE rule S (X ).. A _
c. Evaluate the mean square error performances of S and S.. 15113.23 We make a noisy measurement 1’ = S +VN of a signal S with pdf'ﬁg in the presence of
' additive noise N having pdf ﬂy, and we know that ﬁs-‘IM-s . n) = 15; (3)13; (n). a. Evaluate the conditional ednyIs-(yls) = P(Y s ylS = s) in terms off; and far.
b. Evaluate fsly in terms off}; and'fm _ - - - '
c.. If 5 ~ N(m, 1) and N ~ N(O,0'2), evaluate the MAP sat). d- Evaluate the MLE Em. - El3.24 A voltage V ~ 8(1) is measured by an instrument having response X that contains an
additive independent normally distributed error of zero mean and variance oz. Evaluate the density 13; (x). Evaluate the MLE for). Evaluate the MAP V(X). . We attempt to infer V from 9'00 = aX -+ b..-What are the best choices of a, b
to yield a minimum mean square estimate? ' 9.0 93's» ...

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- Spring '05
- HAAS
- Statistics, Probability, error probability, mle