By Linda J. S. Allen

An advent to Stochastic tactics with functions to Biology, moment Edition offers the elemental conception of stochastic techniques priceless in figuring out and using stochastic how to organic difficulties in components akin to inhabitants progress and extinction, drug kinetics, two-species festival and predation, the unfold of epidemics, and the genetics of inbreeding. due to their wealthy constitution, the textual content specializes in discrete and non-stop time Markov chains and non-stop time and country Markov processes.

New to the second one Edition

  • A new bankruptcy on stochastic differential equations that extends the fundamental concept to multivariate procedures, together with multivariate ahead and backward Kolmogorov differential equations and the multivariate Itô’s formula
  • The inclusion of examples and workouts from mobile and molecular biology
  • Double the variety of workouts and MATLAB® courses on the finish of every chapter
  • Answers and tricks to chose workouts within the appendix
  • Additional references from the literature

This version keeps to supply a good advent to the basic idea of stochastic procedures, besides quite a lot of functions from the organic sciences. to raised visualize the dynamics of stochastic tactics, MATLAB courses are supplied within the bankruptcy appendices.

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Extra info for An Introduction to Stochastic Processes with Applications to Biology, Second Edition

Example text

These formulas are summarized below. µX = PX (1) = MX (0) = KX (0) and 2 σX   2    PX (1) + PX (1) − [PX (1)] = M (0) − [M (0)]2 X X     KX (0) 22 An Introduction to Stochastic Processes with Applications to Biology Generating functions for linear combinations of independent random variables can be defined in terms of the generating functions of the individual random variables. Suppose X1 , X2 , . . , Xn are n independent random variables and Y is a linear combination of these random variables, n Y = ai Xi .

The Markov inequality states that for a nonnegative random variable X with expectation µ, µ Prob{X ≥ c} ≤ , c > 0. c Review of Probability Theory 39 31. Another well-known inequality in probability theory is Chebyshev’s inequality. The Chebyshev inequality states that for any random variable X with finite expectation µ and positive variance σ 2 , Prob{|X − µ| < kσ} ≥ 1 − 1 , k > 0. k2 Use the Markov inequality in Exercise 30 with Y 2 = (X − µ)2 and E(Y 2 ) = σ 2 to verify the Chebyshev inequality.

Xf (x) dx = 0 = −∞ −∞ In addition, ∞ ∞ x2 exp −x2 /2σ 2 √ dx σ 2π −∞ −∞ ∞ ∞ σx exp(−x2 /2σ 2 ) σ exp(−x2 /2σ 2 ) √ √ =− + dx 2π 2π −∞ −∞ E(X 2 ) = x2 f (x) dx = ∞ = σ2 f (x) dx = σ 2 , −∞ where integration by parts is used in the first integral, u = x, and dv = xf (x) dx. Therefore, the normal distribution N (0, σ 2 ) has mean and variance, 2 µX = 0 and σX = σ 2 . In a similar manner, it can be shown that the normal 2 distribution N (µ, σ 2 ) has mean and variance, µX = µ and σX = σ 2 . The fourth moment is computed using the same technique, ∞ ∞ x4 exp −x2 /2σ 2 √ dx σ 2π −∞ −∞ ∞ ∞ σx3 exp(−x2 /2σ 2 ) 3σx2 exp(−x2 /2σ 2 ) √ √ =− + dx 2π 2π −∞ −∞ E(X 4 ) = x4 f (x) dx = ∞ = 3σ 2 x2 f (x) dx = 3σ 2 E(X 2 ) = 3(σ 2 )2 .

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