Conditional Distribution for Two Random Variables
Independence for Two Random Variables
Functions of Two Random Variables
ECE 2521: Analysis of Stochastic Processes
Lecture 7
Department of Electrical and Computer Engineering
University of Pittsburgh
October,
10
th
2018
Mircea Lupu, PhD
Lecture 7
ECE 2521: Analysis of Stochastic Processes
Conditional Distribution for Two Random Variables
Independence for Two Random Variables
Functions of Two Random Variables
Conditioning by an Event
Conditioning by a Random Variable
Law of Iterated Expectations (Total Expectation Theorem)
Conditioning by an Event
We consider the probability model for two or more random
variables given the knowledge that some event
B
has occured
Conditional Joint Probability Mass Function
For two discrete random variables
X
and
Y
and a conditioning
event
B
(Prob(
B
)
>
0) the conditional joint PMF of
X
and
Y
:
p
X
,
Y

B
(
x
)
=
Prob(
X
=
x
and
Y
=
y

B
)
=
(
p
X
,
Y
(
x
,
y
)
Prob(
B
)
if (
x
,
y
)
∈
B
0
otherwise
.
The conditional joint PMF is nonzero for a pair (
x
,
y
), if (
x
,
y
)
is contained in the conditioning event
B
, and zero otherwise
Satisfies the axioms of probability:
(1)
Nonnegativity:
p
X
,
Y

B
(
x
,
y
)
≥
0.
(2)
Normalization:
∑
(
x
,
y
)
∑
∈
B
p
X
,
Y

B
(
x
,
y
) = 1
Lecture 7
ECE 2521: Analysis of Stochastic Processes
Conditional Distribution for Two Random Variables
Independence for Two Random Variables
Functions of Two Random Variables
Conditioning by an Event
Conditioning by a Random Variable
Law of Iterated Expectations (Total Expectation Theorem)
Conditioning by an Event
Conditional Joint Probability Density Function
For two continuous random variable
X
and
Y
and a
conditioning event
B
(Prob(
B
)
>
0) the conditional joint PDF
of
X
and
Y
:
f
X
,
Y

B
(
x
,
y
) =
(
f
X
,
Y
(
x
,
y
)
Prob(
B
)
if (
x
,
y
)
∈
B
0
otherwise
.
Satisfies the axioms of probability:
(1)
Nonnegativity:
f
X
,
Y

B
(
x
,
y
)
≥
0.
(2)
Normalization:
R
∞
∞
R
∞
∞
f
X
,
Y

B
(
x
,
y
)
dxdy
= 1
Lecture 7
ECE 2521: Analysis of Stochastic Processes
Conditional Distribution for Two Random Variables
Independence for Two Random Variables
Functions of Two Random Variables
Conditioning by an Event
Conditioning by a Random Variable
Law of Iterated Expectations (Total Expectation Theorem)
Conditional Expectations
The
conditional expected value
of a function of two
random variables
W
=
g
(
X
,
Y
) given condition
B
is:
E
[
W

B
]
=
E
[
g
(
X
,
Y
)

B
]
=
∑
(
x
,
y
)
∑
∈
B
g
(
x
,
y
)
p
X
,
Y

B
(
x
,
y
)
if
X
,
Y
discrete
R
∞
∞
R
∞
∞
g
(
x
,
y
)
f
X
,
Y

B
(
x
,
y
)
dxdy
if
X
,
Y
continuous
Then
conditional variance
of
W
is computed by:
Var
[
W

B
] =
E
[
W
2

B
]

(
E
[
W

B
])
2
.
Lecture 7
ECE 2521: Analysis of Stochastic Processes
Conditional Distribution for Two Random Variables
Independence for Two Random Variables
Functions of Two Random Variables
Conditioning by an Event
Conditioning by a Random Variable
Law of Iterated Expectations (Total Expectation Theorem)
Example 1 (Discrete)
Random variables
L
and
T
have the following joint PMF:
P
L
,
T
(
l
,
t
)
t
= 40 sec
t
= 60 sec
l
= 1 page
0.15
0.1
l
= 2 pages
0.3
0.2
l
= 3 pages
0.15
0.1
For the random variable
V
=
LT
, we define the event:
A
=
{
V
>
90
}
Find the following:
(1)
the conditional PMF
p
L
,
T

A
(
l
,
t
)
(2)
E
[
V

A
]
(3)
Var[
V

A
].
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 Probability theory, conditional distribution, continuous random variable X, X given Y, Analysis of Stochastic Processes