Stochastic Signals and Systems
Random Variables
Virginia Tech
Fall 2008
Functions of a Random Variable
We treat the problem of calculating the probability density
function and the cumulative distribution function of a random
variable
Y
that is a given function
g
(
X
)
of a random variable
X
having a known probability density function
f
X
(
x
)
.
1
Y
is also a random variable.
2
The probabilities with which
Y
takes on various values
depend on the function
g
(
x
)
and on
f
X
(
x
)
.
3
We assume the function
g
(
x
)
has a unique value for all
values of
x
.
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Discrete Random Variable
The problem is simplest when the input random variable
X
is of
discrete type, taking values in a countable set
x
1
,
x
2
,
. . .
with
probabilities
P
[
X
=
x
k
]
,
k
=
1
,
2
, . . .
which sum to 1. Then the
random variable
Y
takes on only a countable set of values
y
1
=
g
(
x
1
)
,
y
2
=
g
(
x
2
);
. . .
y
k
=
g
(
x
k
);
. . .
If
y
=
g
(
x
)
is a monotone function of
x
, it has a unique inverse,
which we write as
x
=
g

1
(
y
)
Then the probabilities of the various values of the random
variable
Y
are simply
P
[
Y
=
y
k
] =
P
[
X
=
g
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 Fall '08
 DASILVER
 Probability theory, probability density function

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