Probability &
Statistics
Conditional
PMF on an
Event
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
1/27
Probability & Statistics
Discrete Random Variables 2
TRAN Hoang Tung
NGUYEN Minh Huong
NGUYEN Mai Chung
University of Science and Technology of Hanoi (USTH)
September 17, 2019
Probability &
Statistics
Conditional
PMF on an
Event
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
2/27
The Course So Far
What we have learnt?
Probability &
Statistics
Conditional
PMF on an
Event
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
2/27
The Course So Far
What we have learnt?
Fundamental probability
Probability &
Statistics
Conditional
PMF on an
Event
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
2/27
The Course So Far
What we have learnt?
Fundamental probability
Discrete random variable, PMF
Probability &
Statistics
Conditional
PMF on an
Event
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
2/27
The Course So Far
What we have learnt?
Fundamental probability
Discrete random variable, PMF
Expectation: mean, variance, standard deviation
Probability &
Statistics
Conditional
PMF on an
Event
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
3/27
1
Conditional PMF on an Event
Conditional PMF
Conditional Expectations
Total Expectation Theorem
2
Multiple Discrete Random Variables
Joint PMF
Marginal PMF
Expectation
3
Conditional PMF on a Random Variable
Conditional PMF
Conditional Expectations
Independence
Probability &
Statistics
Conditional
PMF on an
Event
Conditional
PMF
Conditional
Expectations
Total
Expectation
Theorem
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
4/27
1
Conditional PMF on an Event
Conditional PMF
Conditional Expectations
Total Expectation Theorem
2
Multiple Discrete Random Variables
3
Conditional PMF on a Random Variable
Probability &
Statistics
Conditional
PMF on an
Event
Conditional
PMF
Conditional
Expectations
Total
Expectation
Theorem
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
5/27
Conditional PMF on an Event
Conditional PMF
The
conditional PMF
of
X
, given an
event
A
with
P
(
A
)
>
0
,
is defined by
p
X

A
(
x
)
,
P
(
X
=
x

A
)
,
P
(
{
X
=
x
} ∩
A
)
P
(
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
5/27
Conditional PMF on an Event
Conditional PMF
The
conditional PMF
of
X
, given an
event
A
with
P
(
A
)
>
0
,
is defined by
p
X

A
(
x
)
,
P
(
X
=
x

A
)
,
P
(
{
X
=
x
} ∩
A
)
P
(
A
)
Sanity check: verify that
X
x
p
X

A
(
x
) = 1
Probability &
Statistics
Conditional
PMF on an
Event
Conditional
PMF
Conditional
Expectations
Total
Expectation
Theorem
Multiple
Discrete
Random
Variables
Conditional
PMF on a
Random
Variable
6/27
Conditional PMF 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
p
X

A
(
x
)
.
Probability &
Statistics
Conditional
PMF on an
Event
Conditional
PMF
Conditional
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 Probability theory, value Y