Download e-book for kindle: An introduction to stochastic processes by D. Kannan

By D. Kannan

ISBN-10: 0444003010

ISBN-13: 9780444003010

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Extra resources for An introduction to stochastic processes

Example text

Method 2 is the way to properly generate points “at random,’’ but it has the flaw of wasting computational effort generating many points that are then rejected for use (the ones that fail the x 2 + y 2 ≤ 1 condition). Method 1 would be so much nicer to use, if we could eliminate the nonuniform clumping effect near the center of the circle. tex Introduction This is, in fact, not hard to do once the reason for the clumping is identified. , inside a circle with area πr 2 . That is, a fraction r of the points fall inside a smaller circle concentric with K, with an area r 2 as large as the area of K.

In general, the probability of getting m correct pairings when assigning M terms to M presidents, where the M terms are each uniquely assigned to a president, M−m (−1)k 1 is given by m! For m = 0 and M = 24 this formula gives the k =0 k! 368. See, for example, Emanuel Parzen, Modern Probability Theory and Its Applications (New York: John Wiley & Sons, 1960, pp. 77–79). You can find a scholarly, readable history of Montmort’s problem, including detailed discussions of how the greats of yesteryear calculated their solutions to the problem, in L.

The analysis in that paper inspired the following related question. If two points are independently and uniformly located in the unit interval, they divide that interval into three segments. What is the probability that those three segments form an obtuse triangle? You can find a theoretical analysis of this question in Mathematics Magazine (November–December 1973, pp. 170558 · · · . 170567 using one million simulations of randomly dividing the unit interval into three parts. m. Lines 03 and 04 define the variables point1 and point2, and lines 05 through 09 ensure that their values are such that 0 < point1 < point2 < 1.

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An introduction to stochastic processes by D. Kannan

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