Stochastic Processes
nctuee07f
Homework 1
Due on Friday,
10/12/2007
,
1:00 PM
at
EC 713
1. Let
X
be a continuous random variable, and
Y
a discrete random variable. Show that the pdf
f
X
(
x
) can be expressed by
f
X
(
x
) =
X
n
f
X

Y
(
x

Y
=
n
)
P
[
Y
=
n
]
.
(Hint: Use
f
X
(
x
) =
d
dx
P
[
X
≤
x
]
.
)
2. A binary signal
S
∈ {
1
,
+1
}
is transmitted, and we are given that
P
(
S
= 1) =
P
(
S
=

1) =
1
/
2. The received signal at the receiver is modeled by
Y
=
S
+
N,
where
N
is Gaussian noise, with zero mean and variance
σ
2
, independent of
S
. What is the
probability that
S
=

1, given that we have observed
Y
=
y
?
3. (a) Show that
E
[
E
[
X

Y
]] =
E
[
X
]
.
(b) Let
X
and
Y
be two random variables, either discrete or continuous. Show that
E
h
g
(
X
)
·
h
(
Y
)
ﬂ
ﬂ
ﬂ
Y
=
y
i
=
h
(
y
)
·
E
h
g
(
X
)
ﬂ
ﬂ
ﬂ
Y
=
y
i
for any realvalued functions
g
and
h
.
(c) The conditional variance of
X
given
Y
=
y
is deﬁned by
Var(
X

Y
=
y
)
,
E
£
(
X

E
[
X

Y
=
y
])
2

Y
=
y
/
,
it is clear that Var(
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 Winter '10
 GFung
 Probability theory, Singular value decomposition, continuous random variable, discrete random variable, UBV

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