I.\ v I b/ J
5.9. Explain why none of the following matrices can be covariance matrices associated
_ with real random vectors
Q1 (2-K vu/ all!
6 1 + j 2
' 5 1
l 6
(C)
5.11. (K. Fukunaga [56, p. 33].) Let K1 and K2 be positive denite covariance matrices
an

3.10. Let X be aCauss-ian r.v. with pdf
2
fx(93)
1
m -~e
x/27r
Let Y r 9(X) where g() is the nonlinear function given as
1, a3<m1
33',. 1S$1,
1, 33>1.
It is called a saturable limiter function.
a) Sketch 9(a); b) Find Fy( )', c) Find and sketch fy( )
3.11

\ _-./.
1.23. Assume there are 3 machines, A, B, and C in a semiconductor manufacturing facility
that make chips. They manufacture, respectively, 25, 35, and 40 percent of the total
semiconductor chips there. Of their outputs, respectively, 5, 4, and 2 pe

Moment-generating functions
Definition: For a R.V. X, its moment generating function
(MGF) is (t is a complex number)
(t ) = E[e ] = etx f X ( x)dx
tX
Note that except for a sign reversal, the MGF is the two-sided
Laplace transform of the pdf
The inve

Chapter 2: Random Variables
Dr. Y.C. Wu
ELEC 8505
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1
Definition of Random Variables
We are interested in converting a physical sample space
into a new one which is numerical
E.g., For tossing a coin, the outcomes are H or T, which
are not numerica

Chapter 1: Introduction
Dr. Y.C. Wu
ELEC 8505
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1
Basic definitions
nE
n n
Frequency of occurrence: P[ E ] lim
Probability based on Axiomatic Theory
Experiment
Sample space : set of all possible outcomes
Events E: a subset of
Example 1: Experi

Power Spectral Density
The power spectral density (PSD) is defined as the FT of
autocorrelation function
S XX ( ) = m = RXX [m]exp( j m)
for
The autocorrelation function can be recovered from PSD
1
RXX [m] = IFT cfw_S XX ( ) = (2 ) S XX ( )e j m d
So,

Chapter 6: Random Sequences
Dr. Y.C. Wu
ELEC 8505
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1
Basic Concepts
Let . We know that X[] a mapping forming a R.V.
Let . Let X[n,] be a mapping of sample space into a space of
complex-valued sequences (n takes on integer values), then X[n,] is
a

Chapter 5: Random Vectors
and Parameter Estimation
Dr. Y.C. Wu
ELEC 8505
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1
Joint Distribution and Densities
Let X = [ X 1 X 2 . X n ]T
Joint PDF: FX ( x) = P[ X 1 x1 ,., X n xn ]
P[ X x]
Xx
Similar to 1D case, FX ( ) = 1,
FX ( ) = 0
n FX (x)
J

Chapter 7: Random Processes
Dr. Y.C. Wu
ELEC 8505
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1
Basic Definitions
For a random sequence, the time
axis is discrete. When the time axis
becomes uncountable, we have a
random process.
Formally, a random process is a
mapping from the sample spac

Chapter 3: Functions of R.V.s
Dr. Y.C. Wu
ELEC 8505
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Question: Given a rule g(.), and a R.V. X with pdf fX(x), what is
the pdf fY(y) of the R.V. Y=g(X) ?
The answers can range from very simple to complicated to
almost impossible
Example: Given

Chapter 4: Expectation and
Introduction to Estimation
Dr. Y.C. Wu
ELEC 8505
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Expectation of a R.V.
In general, a R.V. is specified by its pdf
Sometimes, we are interested in summarizing certain
properties of a R.V. by a few numbers
E.g., sample