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As in classical probability theory, the probability density corresponding to the distribution F (x; t) is given by, f (x; t) = ∂F (x; t) . 4) and so on for higher-order joint distributions. In general, F (x1 , . . , xn ; t1 , . . , tn ) = P {ξ(t1 ) ≤ x1 , . . 5) n f (x1 , . . , xn ; t1 , . . , tn ) = ∂ F (x1 , . . , xn ; t1 , . . , tn ) . ∂x1 . . 6) Note that the nth -order distribution function determines all lower-order distribution functions, and in fact, the nth -order distribution function completely determines the stochastic process.

47) m=−∞ with z a complex number, and the integral the unit circle |z| = 1 in the complex plane. running along (c) Derive a differential equation for the probability-generating function and show that its solution, subject to the appropriate initial condition, is given by Q(z, t) = exp z + z −1 − 2 λt . (d) Show that the solution of the equation from part 7a is given by, pm (t) = e−2λt I|m| (2λt), where In (x) is a modified-Bessel function of integer order n. (e) Deduce that the leading term of the asymptotic series for the probability density function from part 7d, when t, m → ∞ with m2 /t fixed, has the form, pm (t) = √ 1 m2 exp − 4λt 4πλt .

6). A fly lands on the web at some vertex. 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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