6.262.Lec13

6.262.Lec13 - DISCRETE STOCHASTIC PROCESSES Lecture 13...

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Lecture 13 - 3/17/2010 1 DISCRETE STOCHASTIC PROCESSES Lecture 13 Ensemble Averages (Expectations ) Review: Results from Section 3.5: Elementary Renewal Theorem Blackwell's Theorem Section 3.6 - Distribution of Age Key Renewal Theorem Joint and Conditional Densities of the Age and Duration
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Lecture 13 - 3/17/2010 2 Elementary Renewal Theorem The following result holds in general. Theorem: Let Nt t af lq ; 0 be a renewal process with the interrenewals having possibly infinite mean X and possibly infinite variance. Let m(t) = E[N(t)]. Then ( ) 1 lim t mt tX →∞ ⎡⎤ = ⎢⎥ ⎣⎦
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Lecture 13 - 3/17/2010 3 . t →∞ Blackwell’s Theorem Simplest View of Blackwell’s Theorem: If has a rational transform (or, more generally, a smooth density), then i) m(t) is differentiable, ii) exists, and iii) This view is too narrow, however, because may take on discrete values, and as a result, m(t) may not be differentiable (more specifically, its derivative may be either 0 or at every point.) The statements below consider the chord, rather than the slope, of m(t) as . Definition : A distribution function is called arithmetic with span d > 0 if all possible values of the random variable are multiples of d and d is the largest such number (i.e., an arithmetic rv is an integer rv with a scale factor.) Blackwell's Theorem (proof in Feller): If a renewal process has non-arithmetic interarrival intervals, then for any 0 δ > ,
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This note was uploaded on 10/21/2010 for the course EE 5581 taught by Professor Moon,j during the Spring '08 term at Minnesota.

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6.262.Lec13 - DISCRETE STOCHASTIC PROCESSES Lecture 13...

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