waitingtime - T E = ∞ = ∞ p n =1(1-p n We want to know...

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Summer 2007 STAT333 Summary of the waiting time random variable r.v. Let E be an event and T E be the waiting time for the 1st E . Note that in general, the possible range of T E is R = { 1 , 2 , 3 ,... } ∪{∞} , where {∞} means we can not observe E . We are interested in: 1. can we Fnally observe E ? 2. if we can Fnally observe E , how long on average are we going to take for observing E ? According to these two points, we can classify r.v. into three types. 1. Proper r.v. : P ( T E < ) = 1 or P ( T E = ) = 0 (a) if E ( T E ) < , then T E is a short proper r.v. . (b) if E ( T E ) = , then T E is a null proper r.v. . 2. improper r.v. : P ( T E < ) < 1 or P ( T E = ) > 0 Consider a speciFc case in which a coin is tossed repeatedly, at the nth toss, p n = P (observe H at toss n ). Let T E be the waiting time for the 1st H . then P
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Unformatted text preview: ( T E = ∞ ) = ∞ p n =1 (1-p n ) . We want to know the above probability is equal to 0 or not. Sum-Product Lemma can help us for the above problem. Sum-Product Lemma : ∞ p n =1 (1-p n ) = 0 ⇔ ∞ s n =1 p n = ∞ or equivalently ∞ p n =1 (1-p n ) > ⇔ ∞ s n =1 p n < ∞ . Understanding Sum-Product Lemma : Let X = the number of H obtained in the sequence, I n be the indicator variable for the event that you observe H at toss n . Then X = ∞ s n =1 I n and E ( X ) = ∞ s n =1 p n . So Sum-Product Lemma tells us we can Fnally observe H if and only if we can observe H inFnite number of times. 1...
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