Name: _________________________ Student ID No: ______________
UNIVERSITY OF CALIFORNIA
College of Engineering
Department of Electrical Engineering and
Computer Sciences
Professor Tse Fall 1998
EECS 126 — MIDTERM #1
2 October 1998, 11:10–12:10
[20 pts.] 1a.Suppose that are events and . What can you say about if:
i) and are independent?
ii) and are mutually exclusive?
iii) ?
iv) ?
[10 pts.] b. If the occurrence of event makes more likely (i.e., ), then does the
occurrence of event make more likely? Justify you answer.
[30 pts.] 2. There are 2 machines having lifetimes distributed with cdf’s and . Suppose one of
the 2 machines is randomly picked with equal probability and put in operation at time 0.
Conditional on the fact that the machine is still running at time , what is the probability
that it is machine 1 that was picked?
[20 pts.] 3a.Consider a binary channel with cross-over probability
Suppose .
Further suppose you use a detection rule which decides that 0 is transmitted if 0 is
received, and 1 is transmitted if 1 is received. Find the probability that you will make an
error.
[20 pts.] b. Suppose now that . A student thinks that a random detection
rule can perform better than the detection rule above. Namely, the student flips a biased
coin with . If the coin lands on a tail, the student decides that what is trans-
mitted is the same as what is received; if the coin lands on a head, he decides that what is
transmitted is opposite to what is received. What is the probability that the student makes
an error using this rule? Is this a better rule than the one in (a)?
E F,
P E( ) 0.4=
P E F( )
E
F
E
F
F E⊂
E F⊂
B
A
P A B( ) P A( )>
A
B
F1
F2
t
Poutput 1 input 0==( ) ε1
=
Poutput 0= input 1=( ) ε2
=
Pinput 0=( ) p=
Pinput 1=( ) 1p–=
p1
2
--- ε1
, ε2 ε1
2
---
<= = =
Phead( ) ε=
