An Introduction to Probability Theory and Its Applications, by William Feller

By William Feller

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Extra resources for An Introduction to Probability Theory and Its Applications, Vol. 2

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It is called the strong Markov property. tions with integrals. For typographical simplicity we let n = 3. As in many similar situations we use a symmetry argument. With probability one, no two among the variables Xi are equal. Neglecting an event of probability zero the six possible orderings of Xh X2, X3 according to magnitude there.. fore iepresent six mutually exclusive events of eqi~al probability. To calculate the distribution of the order statistics it suffices therefore to consider the contingency· Xl < X2 < X3 • Thus .

It follows' that, on the average, in one out of two cases either Paul or Peter,has reason for complaint. The obsened frequency incn~ases in practice because very short waiting times- naturally pass unnoticed. , (c) Parallel waiting lines. ). jne I have to stay in it. Mr. Smith, who drove-behind me, occupies the place that 1 mjght have chosen and I keep watching whether he is ahead of or behind 'me. Most of the time we stand still, but occasionally one'line or the other moves one car-length, forward.

We apply this notion to the particular case of independent random variables Xl' .... x. Examples. (a) Parallel waiting lines. Interpret Xh • . ,X n as the lengths of n service times commencing at epoch 0 at a post office with n counters. The order statistics repre~ent the successive epochs of terminatio,ns or, as one might say, the epochs of the successive discharges (the,"output process"). In particular, X(l) is the waiting time for the first discharge. Now if the assumed lack of aftereffect is meaningful, the waiting time X(l) must have the Markov property, that is, X(l) must be exponentially distributed.

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