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Remarks: 1. The distribution function Eq. 5 is not arbitrary, but must satisfy the following, ∙ Symmetry condition. For every permutation (????1 , ????2 , . . , ???????? ) of (1, 2, . . , ????), we must have ???? (????????1 , ????????2 , . . , ???????????? ; ????????1 , ????????2 , . . , ???????????? ) = ???? (????1 , ????2 , . . , ???????? ; ????1 , ????2 , . . , ???????? ). This condition follows from the AND logic of the distribution function: the statements ‘A AND B AND C’ and ‘A AND C AND B’ are equivalent. ∙ Compatability condition. For ???? < ????, we must have ???? (????1 , .

Guess first, then run Monte Carlo simulations to make a table of ????2????,2???? for ???? = 10 and ???? = 0, 1, 2, . . , 10. Follow the convention that in the event of a tie, the previous leader retains the lead. CHAPTER 2 RANDOM PROCESSES By the time the Ornstein and Uhlenbeck paper appeared, it had become clear that Brownian motion was a phenomenon that could not be handled in a rigorous fashion by classical probability theory. For one thing, although Brownian motion is certainly a random affair, the variables that describe it are not the ordinary random variables of classical probability theory, but rather they are random functions.

I. Suppose the fly dies of fright, and the spider moves blindly along the web to find it. On average, how many steps must the spider take in order to find the fly? How does that number change as the number of radial vertices increases? (In Fig. ) Can you imagine a mapping of this problem to a one-dimensional random walk? Verify your simulation results with an analytic estimate. ii. Suppose the web is very dusty so that the fly can easily move along every strand (for simplicity, assume the time to traverse along a strand is independent of the strand length).

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Applied stochastic processes in science and engineering by Scott M.

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