| 1. |
(20%)Let A ans B be nxn matrices.
Then, A is similar to B if, for some nonsingular P,
P-1AP=B. Let A be similar to B.
Prove that, for any positive integer n, An
is similar to Bn. |
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| 2. |
(20%)Let X1,X2,...,Xn
be independent and identically distributed (iid) random variables haveing
variance  , where n is a positive
and known integer. (a) show that  ,
where Cov(A,B) represents the covariance between A and B,
c is a particular constant, and  .
(b) What can you say about the independence between 
based on the result given in Part(a)? |
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| 3. |
(20%)Let K1,K2,...,Kn
be independent Poisson random variables such that E[Ki]= i. |
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(a) |
Find the moment generating function (MGF) of W=K1+K2+...+Kn. |
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(b) |
Does W have a Gaussian distribution? Why? Answer this
question based on the MGF that you obtained in Part(a). |
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4. |
(20%)Let  .
Find a simple numerical value for  .
(Hint:Consider the diagonalization problems for A and A43,
respectively, and find a link between these two problems.) |
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| 5. |
(20%)You are asked to estimate a random variable Y
from multiple related observations or random variables X1,X2,...,XN,
which can be considered to be components of a random vector X. The
random variables may be complex in general, and an estimate is sought in
the form  , where A=[a1
a2 ... aN]T, mx=E[X],my=E[Y]
and where ai are in general complex coefficients chosen to minimize
the mean square error  . |
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(a) |
Show that CxA=cXY
, where
cXY
= E[(X-mx)(Y-mY)*]
and Cx
= E[(X-mx)(X-mx
)*T]. |
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(b) |
Find the minimum mean square error for the following data:
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