STAT 321 - Stochastic Signals and Systems
Assignment 4
Due: Wednesday, December 7 at 9:00pm.
Submit your assignment online in the pdf format under module \Assignments". You can either
typeset your solutions or scan a handwritten copy.
Assignments are to be completed individually.
Dene notation for events and random variables, and include all steps of your derivations. Writing
down the nal answer will not be sucient to receive full marks.
Please make sure your submission is clear and neat. It is a student's responsibility to ensure that
the submitted le is in good order (e.g., not corrupted and contains what you intend to submit).
Late submission penalty: 1% per hour or fraction of an hour. (In the event of technical issues
with submission, you can email your assignment to the instructor to get a time stamp but submit
on Canvas as soon as it becomes possible to make it available for grading.)
1. We have 3 coins which we will refer to as A, B and C. They look identical, but they have dierent
biases. For Coin A, P (H) = 0:3; P (T ) = 0:7; for Coin B, P (H) = 0:7; P (T ) = 0:3 and for Coin C,
P (H) = 0:5; P (T ) = 0:5.
(a) Alice chooses one of the 3 coins completely at random. She tosses the coin 2 times; the result
of the rst toss is a `H' whereas the result of the second toss is a `T'. Which coin, A, B or C,
should Alice guess she chose if she wishes to minimize the conditional probability of error,
PejHT ?
(b) Under a dierent scenario, Bob gives Alice a coin which he selected with the following
probabilities: P (A) = 0:6; P (B) = 0:4 and P (C) = 0. Alice knows these a priori probabilities.
She tosses the coin once. The possible outcome is a `H' or a `T'. For each possible outcome,
determine the MAP decision rule for Alice to decide which coin, A, B or C, she was given.
What is the resulting average probability of error, Pe?
2. A signal S is equally likely to take on one of 3 possible values: 1; 0;+1. It is sent over an additive
Laplacian noise channel where the pdf of the noise W is
fW (w) =
1
2c
e
jwj
c
The output of the channel is Y = S +W , where S and W are independent.
(a) Determine fY jS(yjs) for s = 1; 0;+1. Sketch your answers (the 3 conditional pdfs) for c = 1.
(b) Determine the minimum Pe decision rule bsMAP (y). Your answer should take the following
form: choose bs = 1 if y is a value in this (or these) interval(s); similarly for bs = 0 and bs = +1.
(c) Determine the resulting average probability of error, Pe, in terms of c.
3. Let S be a positive rv with mean and variance 2. Our goal is to estimate S based on an
observation X of the form X =
p
S W . We can assume that S and W are independent and W has
mean 0 and variance 1.
(a) Determine the linear LMS (LLMS) estimator of S given X = x.
(b) Suppose that Y = X2. Determine the LLMS estimator of S given Y = y. Your answer may
include the fourth moment of W , i.e. E(W 4).
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4. A signal S is equally likely to take on one of 2 possible values: 1;+1. It is sent over an additive
noise channel where noise W U(2;+2). The output of the channel is Y = S +W , where S and
W are independent.
(a) Determine the LMS estimate of S given Y .
(b) Determine the overall average MSE for the estimator in Part (a).
(c) Suppose a MAP decision rule is used to decide whether S = 1 or S = +1. Determine the
MAP decision rule and the resulting overall average probability of error.
(d) Compare the MSEs for the LMS estimator and MAP decoder.
5. (a) Derive the distribution of the number of Bernoulli trials required to get r successes.
(b) Suppose that the probability of success is p = 0:1. Sketch the distributions of the number of
Bernoulli trials required to get r = 1 and r = 2 successes.
6. Suppose that earthquakes occur according to a Poisson process with rate = 2 and the time unit is
a month. Thus, the rate of occurrence is 2 per month.
(a) Determine the probability that at least 2 earthquakes occur in the next 3 months.
(b) Determine the probability distribution of the time from now until the next earthquake.
7. The autocorrelation function of a random sequence fAng is dened as
Ra(m)
4
= E(AnAn+m):
Suppose that the elements of the sequence fAng1n=1 are independent, identically distributed
random variables taking on one of three possible values f1; 0;+2g with probabilities P (1) = 14 ,
P (0) = 14 and P (+2) =
1
2 . Determine Ra(m).