Covariance-6

Covariance-6 - Indicators: Now With Pair-wise Flavor!...

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1 Indicators: Now With Pair-wise 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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Covariance-6 - Indicators: Now With Pair-wise Flavor!...

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