Expectations
Total
Expectation
Theorem
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
7/27
Conditional PMF on an Event
Probability &
Statistics
Conditional
PMF on an
Event
Conditional
PMF
Conditional
Expectations
Total
Expectation
Theorem
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
8/27
Conditional Expectations on an Event
Conditional Expectation
The
conditional expectation
of
X
given an event
A
with
P
(
A
)
>
0
, is defined by
E
[
X

A
]
,
X
x
xp
X

A
(
x
)
Probability &
Statistics
Conditional
PMF on an
Event
Conditional
PMF
Conditional
Expectations
Total
Expectation
Theorem
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
8/27
Conditional Expectations on an Event
Conditional Expectation
The
conditional expectation
of
X
given an event
A
with
P
(
A
)
>
0
, is defined by
E
[
X

A
]
,
X
x
xp
X

A
(
x
)
Expected Value Rule
For a function
g
(
X
)
E
[
g
(
X
)

A
] =
X
x
g
(
x
)
p
X

A
(
x
)
Probability &
Statistics
Conditional
PMF on an
Event
Conditional
PMF
Conditional
Expectations
Total
Expectation
Theorem
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
9/27
Conditional Expectations on an Event
Example
Let
X
be the roll of a fair sixsided die
A
be the event that the roll is an even number
Calculate
E
[
X
]
and
E
[
X

A
]
.
Probability &
Statistics
Conditional
PMF on an
Event
Conditional
PMF
Conditional
Expectations
Total
Expectation
Theorem
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
10/27
Review: Total Probability Theorem
Partition into
A
1
, A
2
, A
3
Have
P
(
A
i
)
for every
i
Have
P
(
B

A
i
)
for every
i
P
(
B
)
=
?
Definition
P
(
B
) =
n
X
i
=1
P
(
A
i
)
P
(
B

A
i
)
Probability &
Statistics
Conditional
PMF on an
Event
Conditional
PMF
Conditional
Expectations
Total
Expectation
Theorem
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
10/27
Review: Total Probability Theorem
Partition into
A
1
, A
2
, A
3
Have
P
(
A
i
)
for every
i
Have
P
(
B

A
i
)
for every
i
P
(
B
)
=
?
Definition
P
(
B
) =
n
X
i
=1
P
(
A
i
)
P
(
B

A
i
)
Definition
Consider event
B
:
X
=
x
p
X
(
x
) =
n
X
i
=1
P
(
A
i
)
p
X

A
i
(
x
)
Probability &
Statistics
Conditional
PMF on an
Event
Conditional
PMF
Conditional
Expectations
Total
Expectation
Theorem
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
11/27
Conditioned on an Event
Total Expectation Theorem
If
A
1
, . . . , A
n
be
disjoint events
that form a partition of the
sample space, with
P
(
A
i
)
>
0
for all
i
, then
E
[
X
] =
n
X
i
=1
P
(
A
i
)
E
[
X

A
i
]
Probability &
Statistics
Conditional
PMF on an
Event
Conditional
PMF
Conditional
Expectations
Total
Expectation
Theorem
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
12/27
Mean and Variance of the Geometric
The Geometric
Geometric with parameter
0
< p
≤
1
p
X
(
k
) = (1

p
)
k

1
p
Probability &
Statistics
Conditional
PMF on an
Event
Conditional
PMF
Conditional
Expectations
Total
Expectation
Theorem
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
12/27
Mean and Variance of the Geometric
The Geometric
Geometric with parameter
0
< p
≤
1
p
X
(
k
) = (1

p
)
k

1
p
Probability &
Statistics
Conditional
PMF on an
Event
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 Fall '14
 LuisDavidGarciaPuente
 Probability theory, value Y