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Elec210 Lecture 8
1
Elec 210: Lecture 8
Conditional Probability Mass Function
Conditional Expected Value
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View Full Document Conditional Probability Mass Function
The effect of partial information about the outcome of a random
experiment on the probability of a discrete random variable is
reflected by the
conditional probability mass function
.
Suppose that we know that an event
C
has occurred (we assume
C
has nonzero probability).
The conditional probability mass
function of
X
given
C
is
By the definition of conditional probability
Elec210 Lecture 8
2
(
)
[

]
X
px
C
P
X
x
C
==
{}
[]
]
[
)

(
C
P
C
x
X
P
C
x
p
X
∩
=
=
Interpretation:
Elec210 Lecture 8
3
S
A
k
x
k
k
x
X
=
)
(
ζ
]
[
)
(
k
k
X
x
X
P
x
p
=
=
Start by recalling
unconditional pmf:
Event:
k
A
∈
Equivalent
Event
]
[
k
A
P
∈
The pmf value
is equivalent to
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View Full Document Interpretation:
Elec210 Lecture 8
4
S
A
k
C
x
k
k
x
X
=
)
(
ζ
Now, with
conditioning…
C
x
X
k
∩
=
)
(
The conditional probability of the
event “
X = x
k
” is the probability of
getting an outcome
ζ
for which
X(
ζ
)=x
k
AND
ζ
is in
C
, normalized by
the probability of
C
occurring.
{}
[]
]
[
)

(
C
P
C
x
X
P
C
x
p
X
∩
=
=
Properties of the Conditional PMF
The conditional pmf has the
same properties as the pmf
.
The pmf is nonnegative:
The values of the pmf sum to 1:
The conditional probability of events B defined by X can be
computed by summing the conditional pmf:
Elec210 Lecture 8
5
(
)
0
X
px
C
≥
[ ]
()
in


where
XX
xB
P
XB
C
p
x
C
B
S
∈
=⊂
all
)
(

)
1
X
k
xS
k
C
pxC
∈
==
C =
x
S
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View Full Document Example: Waiting for Bus
A person arrives at a bus stop at time
ζ
(minutes), taking values in [1…60]
with equal probability
There are 12 buses, with bus
n
arriving at time 5
n
; i.e., the person takes bus 1
if arriving at time {1,2,3,4,5}, bus 2 if arriving at time {6,7,8,9,10}, etc.
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This note was uploaded on 03/27/2011 for the course ELEC 202 taught by Professor ? during the Spring '11 term at HKUST.
 Spring '11
 ?

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