ECE 804, Random Signal Analysis
Oct. 12, 2009
OSU, Autumn 2009
Due: Oct. 19, 2009
Problem Set 3
Problem 1
We consider a random selection of coins, where the probability of heads,
C
for the coins
is a random variable whose pdf is
f
C
(
c
) =
k
·
c
for 0
≤
c
≤
1 and zero otherwise.
(a) Find
k
.
(b) Find
P
(
H
) for a single coin toss.
(c) Let
E
be the event that we choose a coin, flip it
n
times, and heads appears
k
of
these times. Find
P
(
E
).
(d) Find the conditional pdf
f
C

E
(
c

E
).
(e) Find the probability that heads appears in the (
n
+ 1)st toss given that
k
heads has
appeared in the first
n
tosses.
Useful fact:
R
1
0
x
m
(1

x
)
n
dx
=
m
!
n
!
(
m
+
n
+1)!
,
m, n
>
0
Problem 2
Consider a random variable
X
that is passed through a system that has gain with clipping,
defined by
y
=
f
(
x
) =
g
·
x,

x
 ≤
a
ga,
x
≥
a

ga,
x
≤ 
a
where
g
and
a
are given positive constants.
(a) Find
F
Y
(
y
) in terms of
F
X
(
x
). Sketch
F
Y
(
y
) for a given generic
F
X
(
x
) (such as a
Gaussian CDF).
(b) Find
f
Y
(
y
) in terms of
f
X
(
x
).
Sketch
f
Y
(
y
) for a given generic
f
X
(
x
) (such as a
Gaussian pdf).
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 Spring '08
 Staff
 Probability theory, Randomness, Cumulative distribution function, CDF, sample cdf

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