Name: _________________________ Student ID No: ______________
UNIVERSITY OF CALIFORNIA
College of Engineering
Department of Electrical Engineering and
Computer Sciences
Professor Tse Fall 1998
EECS 126 — MIDTERM #2
6 November 1998, 11:10–12:10
[20 pts.] 1a. are two jointly distributed random variables. Are the following statements true in
general? If so, explain. If not, give examples for which they are true.
i)
ii)
iii)
iv)
[15 pts.] 1b: i) If and are independent, is ? Explain.
ii) Suppose now
Is ? Is independent?
[30 pts.] 2. Starting from the origin, a particle takes a random walk in the plane. Each step is of unit
length and is equally likely in any direction, and the direction taken in each step is inde-
pendent of any other. Find the expected squared distance from the origin after steps.
[35 pts.] 3. A message containing a random number bits is transmitted through a binary symmet-
ric channel with cross-over probability . is geometrically distributed with pariame-
ter (i.e., ). Let be the total number of bits
transmitted through correctly. Find:
a. (10 pts.)
b. (15 pts.)
c. pmf of (10 pts.)
(The mean and variance of are and , respectively.)
XY
,
EX Y+
()
EX
()
EY
()
+=
EXY() EX()EY()⋅=
EX
Y
---
⎝⎠
⎛⎞ EX()
EY()
------------=
EXYY 3=()3EX()=
X
Y
EYX
()
EY
()
=
Y
Xwith probability 1
2
---
X– with probability 1
2
---
⎩
⎪
⎪
⎨
⎪
⎪
⎧
=
EYX()EY()=
XY
,
n
M
ε
M
p
PM k=()1p–()
k
p k
0
≥,=
N
EN
()
Var N
()
N
M
1p–
p
-
-------
----
1p–
p2
-
-------
----