MIT6_262S11_lec12

# At trial 3 if j 1 2 x3 is observed and a decision

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Unformatted text preview: rocess, but rather than investigat­ ing the interval (0, t] for a ﬁxed t, we want to inves­ tigate (0, t] where t is selected by the sample path up until t. It is somewhat tricky to formalize this, since t b e­ comes a rv which is a function of {X (t); τ ≤ t}. This approach seems circular, so we have to b e careful. We consider only discrete-time processes {Xi; i ≥ 1}. 10 Let J b e a p ositive integer rv that describes when a sequence X1, X2, . . . , is to b e stopped. At trial 1, X1(ω ) is observed and a decision is made, based on X1(ω ), whether or not to stop. If we stop, J (ω ) = 1 At trial 2 (if J (ω ) ￿= 1), X2(ω ) is observed and a decision is made, based on X1(ω ), X2(ω ), whether or not to stop. If we stop, J (ω ) = 2. At trial 3 (if J (ω ) ￿= 1, 2), X3(ω ) is observed and a decision is made, based on X1(ω ), X2(ω ), X3(ω ), whether or not to stop. If we stop, J (ω ) = 3, etc. At each trial n (if stopping has not yet o ccurred), Xn is observed and a decision (based on X1 . . . , Xn) is made; if we stop, then J (ω ) = n. 11 Def: A stopping trial (or stopping time) J for {Xn; n ≥ 1}, is a p ositive integer-valued rv such that for each n ≥ 1, the indicator rv I{J =n} is a function of {X1, X2, . . . , Xn}. A p ossibly defective stopping trial is the same ex­ cept that J might b e defective. We visualize ‘conducting’ successive trials X1, X2, . . . , until some n at which the event {J = n} o ccurs; fur­ ther trials then cease. It is simpler conceptually to visualize stopping the observation of trials after the stopping trial, but continuing to conduct trials. Since J is a (possibly defective) rv, the events {J = 1}, {J = 2}, . . . are disjoint. 12 Example 1: A gambler goes to a casino and gambles until broke. Example 2: Flip a coin until 10 successive heads appear. Example 3: Test an hypothesis with repeated tri­ als until one or the other hypothesis is suﬃciently probable a p osteriori. Example 4: Observe successive renewals in a re­ newal process until Sn ≥ 100. 13 Suppose...
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## This note was uploaded on 01/13/2012 for the course ELECTRICAL 6.262 taught by Professor Staff during the Fall '11 term at MIT.

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