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By Don S. Lemons

A textbook for physics and engineering scholars that recasts foundational difficulties in classical physics into the language of random variables. It develops the innovations of statistical independence, anticipated values, the algebra of ordinary variables, the crucial restrict theorem, and Wiener and Ornstein-Uhlenbeck techniques. solutions are supplied for a few difficulties.

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Extra resources for An introduction to stochastic processes in physics, containing On the theory of Brownian notion

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Although uniform variables U (m, a) and most random variables do not sum to their own kind, Cauchy variables C(m, a) do so. 2, Adding Uniform Variables. 3) and C1 (m 1 , a1 ) + C2 (m 2 , a2 ) = C(m 1 + m 2 , a1 + a2 ). 4) The latter requires that C1 (m 1 , a1 ) and C2 (m 2 , a2 ) be statistically independent. 4) with moment-generating functions. 4) exploits the so-called random variable transform theorem (Gillespie 1992). 3 Jointly Normal Variables We can make an even more powerful statement: statistically dependent normals, if jointly normal, also sum to a normal.

3) and, as we shall see, encourages us to adopt the sub- and superscripts placed on the unit normal symbol Ntt+dt (0, 1). Because of time-domain continuity, X (t + dt/2) exists and we can formally divide the process-variable increment X (t +dt)− X (t) into the sum of two subincrements so that X (t +dt)− X (t) = [X (t +dt)− X (t +dt/2)]+[X (t +dt/2)− X (t)]. 6) seems trivial, it has a surprising consequence. 7) a condition that has been called self-consistency (Gillespie 1996). Self-consist+ dt2 tency obtains only when the unit normals N t+dt dt (0, 1) and Nt t+ 2 (0, 1) are sta- tistically independent.

When normalized (on the semiinfinite line x ≥ 0), the intensity of surviving photons becomes the photon probability density p(x) = λe−λx x ≥0 = 0 x < 0. 5. Single-slit diffraction. PROBLEMS 31 The random variable so defined is called the exponential random variable E(λ). a. Show that mean{E(λ)} = 1/λ. b. Find the moment-generating function M E (t) of E(λ) for t < λ. c. Use the moment-generating function to find var{E(λ)}. d. Also, find E(λ)n for arbitrary integer n. 4. Poisson Random Variable. The probability that n identical outcomes are realized in a very large set of statistically independent and identically distributed random variables when a each outcome is extremely improbable is described by the Poisson probability distribution Pn = e−µ µn , n!

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