This preview shows pages 1–3. Sign up to view the full content.
Stochastic Signals and Systems
Random Variables
Virginia Tech
Fall 2008
ManyOne Transformations
If several values of
x
can be transformed into one value of
y
,
we say that
y
=
g
(
x
)
represents a manyone value. Thus with
Y
=
X
2
, both
x
and

x
go into the same value of
y
. It is merely
necessary to add the probabilities of all the “inﬁnitesimal
events”
(
x
<
X
<
x
+
dx
)
that go into the same event
(
y
<
Y
<
y
+
dy
)
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentManyOne Transformations
The event
(
y
<
Y
<
y
+
dy
)
occurs when any one of the three
events involving
X
occurs. These events will be mutually
exclusive if
dy
is sufﬁciently small. Hence
P
[
y
<
Y
<
y
+
dy
] =
P
[
x
1
<
X
<
x
1
+
dx
1
] +
P
[
x
2
<
X
<
x
2
+
dx
2
]
+
P
[
x
3
<
X
<
x
3
+
dx
3
]
and in terms of the pdf of
X
and
Y
,
f
Y
(
y
)

dy

=
f
X
(
x
1
)

dx
1

+
f
X
(
x
2
)

dx
2

+
f
X
(
x
3
)

dx
3

.
Dividing by

dy

, we ﬁnd the pdf of
Y
f
Y
(
y
) =
f
X
(
x
1
)
±
±
±
±
dx
1
dy
±
±
±
±
±
±
±
±
x
1
=
g

1
(
y
)
+
...
+
f
X
(
x
k
)
±
±
±
±
dx
k
dy
±
±
±
±
±
±
±
±
x
k
=
g

1
(
y
)
Example
Let
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '08
 DASILVER

Click to edit the document details