ECE 804, Random Signal Analysis
Oct. 11, 2010
OSU, Autumn 2010
Due: Oct. 20
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).
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '05
 UYSALBIYIKOGLU
 Probability theory, Randomness, Cumulative distribution function, CDF, sample cdf

Click to edit the document details