ECE 6010: Lecture 1 – Introduction; Review of Random Variables
5
2
Is independent the same as disjoint?
Note: For
P
(
B
) >
0, if
A
and
B
are independentthen
P
(
A B
)
P
(
A
)
. (Since they are
independent,
B
can provide no information about
A
, so the probability remains unchanged.
If
P
(
B
)
0, then
B
is independent of
A
for any other event
A
F
. (Why?).
Definition 5
A
1
, . . . ,
A
n
F
are
independent
if for each
k
2
, . . . ,
n
and each subset
i
1
, . . . ,
i
k
of 1
, . . .,
n
,
P
(
k
j
1
A
i
j
)
k
j
1
P
(
A
i
j
).
2
Example 7
Take
n
3. Independent if:
P
(
A
1
A
2
)
P
(
A
1
)
P
(
A
2
)
P
(
A
1
A
3
)
P
(
A
1
)
P
(
A
3
)
P
(
A
2
A
3
)
P
(
A
2
)
P
(
A
3
)
and
P
(
A
1
A
2
A
3
)
P
(
A
1
)
P
(
A
2
)
P
(
A
3
).
2
The next idea is important in a lot of practical problem of engineering interest.
Definition 6
A
1
and
A
2
are
conditionally independent
given
B
F
if
P
(
A
1
A
2
B
)
P
(
A
1
B
)
P
(
A
2
B
)
2
(draw picture to illustrate the idea).
Random variables
Up to this point, the outcomes in
could be anything: they could be elephants, computers,
or mitochondria, since
is simple expressed in terms of sets. But we frequently deal with
numbers
, and want to describe events associated with sets of numbers. This leads to the
idea of a random variable.
Definition 7
Given a probability space
(
,
F
,
P
)
, a
random variable
is a function
X
mapping
to
R
. (That is,
X
:
R
), such that for each
a
R
,
ω
:
X
(ω)
a
F
.
2
A function
X
:
R
such that
w
:
X
(ω)
a
F
, that is, such that the events
involved are in
F
, is said to be
measurable
with respect to
F
. That is,
F
is divided into
sufficiently small pieces that the events in it can describe all of the sets associated with
X
.
Example 8
Let
1
,
2
,
3
,
4
,
5
,
6 ,
F
1
,
2
,
3
,
4
,
5
,
6
,
,
. Define
X
(ω)
0
ω
odd
1
ω
otherwise
Then for
X
to be a random variable, we must have
ω
X
(ω)
a