1
Indicators: Now With Pairwise Flavor!
•
Recall
I
i
is indicator variable for event
A
i
when:
Let X = # of events that occur:
•
Now consider pair of events
A
i
A
j
occurring
I
i
I
j
= 1 if both events
A
i
and
A
j
occur, 0 otherwise
Number of pairs of events that occur is
otherwise
0
occurs
if
1
i
i
A
I
n
i
i
n
i
i
n
i
i
A
P
I
E
I
E
X
E
1
1
1
)
(
]
[
]
[
n
i
i
I
X
1
j
i
j
i
I
I
X
2
From Event Pairs to Variance
•
Expected number of pairs of events:
•
Recall: Var(X) = E[X
2
] – (E[X])
2
j
i
j
i
j
i
j
i
j
i
j
i
A
A
P
I
I
E
I
I
E
E
X
)
(
]
[
2
j
i
j
i
A
A
P
X
E
X
E
E
X
X
)
(
])
[
]
[
(
2
2
1
2
)
1
(
]
[
)
(
2
]
[
)
(
2
]
[
]
[
2
2
X
E
A
A
P
X
E
A
A
P
X
E
X
E
j
i
j
i
j
i
j
i
2
])
[
(
]
[
)
(
2
)
(
Var
X
E
X
E
A
A
P
X
j
i
j
i
2
1
1
)
(
)
(
)
(
2
n
i
i
n
i
i
j
i
j
i
A
P
A
P
A
A
P
Let’s Try It With the Binomial
•
X ~ Bin(n, p)
Each trial: X
i
~ Ber(p)
Let event A
i
= trial
i
is success (i.e., X
i
= 1)
2
2
2
2
)
(
]
[
p
p
A
A
P
X
X
E
E
n
X
j
i
j
i
j
i
j
i
j
i
np
A
P
X
E
n
i
i
1
)
(
]
[
p
X
E
i
]
[
2
2
2
)
1
(
])
[
]
[
(
2
1
2
)
1
(
p
n
n
X
E
X
E
E
X
X
2
2
2
2
])
[
(
]
[
])
[
]
[
(
])
[
(
]
[
)
(
Var
X
E
X
E
X
E
X
E
X
E
X
E
X
2
2
2
2
2
2
2
)
(
)
1
(
p
n
np
np
p
n
np
np
p
n
n
)
1
(
p
np
Computer Cluster Utilization
•
Computer cluster with N servers
Requests independently go to server
i
with probability
p
i
Let event A
i
= server
i
receives no requests
X = # of events A
1
, A
2
, … A
n
that occur
Y = # servers that receive ≥ 1 request = N – X
E[Y] after first
n
requests?
Since requests independent:
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 Spring '09
 Variance, Probability theory, var

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