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)
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)
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
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
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