Unformatted text preview: andom variable
and Y knowledge of Y PROCESSES
X (and vice versa). Suppose weProblem Set X based on knowledge of Y using an aﬃne estimator
want to estimate No. 2 Solutions
Fallˆ 2001 + b
X = aY
Issued:
for some a, b Wednesday,to be 12, 2001
constants Sep. determined. Due: Friday, Sep. 21, 2001 Problem 2.1 (a) Find expressions for a,variables X and E [X ] = jointE [Y ] = function is ,given , CY X , Cx,Y ).(where CAB = E [(A −
b in terms of Y whose µX , density µY , CXX CXY by p ( Y y For each of
Consider the random
X,Y
ˆ
E [AtheB − E [Bchoices of thethe mean squared error Ebelow determineminimized. and Y are uncorrelated,)( possible ])]) so that joint density function given [(X − X )2 ] is i) whether X
ii) whether X and Y are independent, and iii) specify the corresponding covariance matrix: (b) For each of the following joint distributions shown below, what are the resulting a, b and estimators? In
ˆ
each case graph the function X on the plot of theσjoint σXY . What observations can you make? When does
density.
XX
ΣXY =
σ
knowledge of Y not aﬀect the aﬃne estimate of X ? XY σY Y
Y
p X ,Y p
Y ( x ,y ) = 2 X ,Y p ( x ,y ) = 1 /2 Y 1 1 X ,Y ( x ,y ) = 1 1
0 1 0 pX,Y (x, y ) = 1 X
1 1 X x, y ≥ 0 &
x+y ≤1 0 otherwise
2 pX,Y (x, y ) = (a) 0...
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This note was uploaded on 01/08/2013 for the course ECE 531 taught by Professor Natasha during the Spring '10 term at Ill. Chicago.
 Spring '10
 Natasha

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