Stochastic

In order for i to be rst at time t ti t and the other

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Unformatted text preview: ore units of time is the same as if we haven’t waited at all.” In symbols P (T > t + s|T > t) = P (T > s) (2.6) To prove this we recall that if B ⇢ A, then P (B |A) = P (B )/P (A), so P (T > t + s|T > t) = P (T > t + s) e (t+s) = =e P (T > t) et s = P (T > s) where in the third step we have used the fact ea+b = ea eb . Exponential races. Let S = exponential( ) and T = exponential(µ) be independent. In order for the minimum of S and T to be larger than t, each of S and T must be larger than t. Using this and independence we have P (min(S, T ) > t) = P (S > t, T > t) = P (S > t)P (T > t) =e t e µt =e (2.7) ( +µ)t That is, min(S, T ) has an exponential distribution with rate + µ. The last calculation extends easily to a sequence of independent random variables T1 , . . . , Tn where Ti = exponential( i ). P (min(T1 , . . . , Tn ) > t) = P (T1 > t, . . . Tn > t) n n Y Y = P (Ti > t) = e i t = e ( 1 +···+ i=1 n )t (2.8) i=1 That is, the minimum, min(T1 , . . . , Tn ), of several independent exponentials has an exponential...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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