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Name: _________________________ Student ID No: ______________
UNIVERSITY OF CALIFORNIA College of Engineering Department of Electrical Engineering and Computer Sciences
Professor Zeitouni Spring 1998
EECS 126 — MIDTERM
April 9, 1998, Thursday, 6-8 p.m.
+ Please set the outline of your solution before carrying out any detailed computations.
[45 pts.] 1. Given the joint pdf of the random vector
Find:
a) the value of.
b)
c). Are independent?
d) the MMSE estimator of given. Compute the resulting mean square error.
Compute:
e).
f).
( X 1 , X 2 )
f (^) X 1 ,^ X 2
( x 1 , x 2 ) k x ( 12 + x 22 ) 0 ≤ x 1 ≤ 1 , 0 ≤ x 2 ≤ 1
0 otherwise
k
FX ( x ) , FY ( y ) , f (^) X ( x ) , f (^) Y ( y )
f (^) X 1 X 2
( x 1 x 2 ) ( X 1 , X 2 )
X 1 X 2
P ( X 1 – X 2 ≥ 1 ⁄ 3 )
P X ( 1 = X 2 )
[20 pts.] 2. A communication channel is defined as follows:
The transmission of the string ‘000’ means message A was transmitted; the transmission of the string ‘111’ means message B was transmitted. Messages A and B are equally likely to be transmitted. The receiver observes the 3 output bits corresponding to the (corrupted output) from the message transmitted.
a) Find.
b) Define the decision rule: decide message A was transmitted if in the output the majority of bits were 0, otherwise decide message B.
Compute.
Channel
input output
P ( output bit ≠input bit) = 0.
P (output string ≠input string)
P ( error)
