Name: _________________________ Student ID No: ______________
UNIVERSITY OF CALIFORNIA
College of Engineering
Department of Electrical Engineering and
Computer Sciences
Professor Ren Fall 1997
EECS 126 — MIDTERM #2
November 17, 1997, Monday 7-9 p.m.
[42 pts.] 1.
Given the joint probability density of two RVs and
a)
Find the value of , and the cdf . (6 pts.)
b)
Find
. (6 pts.)
c)
Find the probability that . (6 pts.)
d)
Find
. (6 pts.)
e)
Find the minimum mean square error estimator of given . Compute the resulting
mean square error. (6 pts.)
f)
Find the linear minimum mean square error estimator of given . Compute the
resulting mean square error. (6 pts.)
g)
Are and independent? Uncorrelated? Orthogonal? Explain your answer. (6 pts.)
[35 pts.] 2.
An electronic system has components. Let the lifetime of each component be
, in hours. Assume that , are mutually independent,
and have identical density . Let the lifetime of the system be .
a)
Suppose the system works only if all components work. Find the pdf and expectation
of . (10 pts.)
b)
Suppose we already know that the system has already lasted 10 hours. Find the condi-
tional pdf and expectation of . (12 pts.)
c)
To increase reliability, we use redundancy by increasing the number of components from
to . Suppose the system works so long as there are at least components working.
Find the cdf of . (13 pts.)
X
Y
fXY x y,( ) k x y+( ) 0x1 0 y1≤ ≤,≤ ≤
0 otherwise
=
k
FXY x y,( )
FXx( ) FYy( ) fXx( ) fYy( ),,,
X Y– 1 2⁄≤
fX Y x y( )
X
Y
X
Y
X
Y
n
Xi i,1 2 …n, , ,=
Xi i,1 2 …n, , ,=
fXix( ) ex– x0≥,=
Y
n
Y
Y
n
2n
n
Y
