By M.I. Zelikin, S.A. Vakhrameev

ISBN-10: 3540667415

ISBN-13: 9783540667414

The single monograph at the subject, this booklet issues geometric equipment within the concept of differential equations with quadratic right-hand aspects, heavily regarding the calculus of adaptations and optimum regulate idea. in response to the author’s lectures, the e-book is addressed to undergraduate and graduate scholars, and clinical researchers.

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Extra info for Control Theory and Optimization I

Example text

X having a Poi(A) distribution, we draw a random sample of X. That is, we take n observations, X i , . . ,Xn^ of X and we assume that the X/^'s have the same distribution function as X and are independent. Next, we write that the estimator A of A (which is the mean of the distribution) is the arithmetic mean of the observations. Similarly, to estimate the mean mx{t) of a stochastic process {X{t),t G T} at time t, we must first take observations X{t,Sk) of the process. p. by the mean of a random sample taken at time t.

Xk\ti,,,. 3) Remark. 3. 1, then we may write, with p := P[{Tails}], that the first-order probability mass function (or probability mass function of order 1) of the process at time n = 2 is given by (2p{l-p)i{x=-0 p2 ifa: = 2 p ( a : ; n - 2 ) - P [ X 2 = x] = <^ ( l - p ) 2 ifx = - 2 0 otherwise First- and second-order m o m e n t s of stochastic processes Just like the means, variances, and covariances enable us to characterize, at least partially, random variables and vectors, we can also characterize a stochastic process with the help of its moments.

S X and Y are independent if and only if their correlation coefficient is equal to zero. An important particular case of transformations of random vectors is the one where the random variable Z := g{Xi,... s X j , . . 106) where the a^'s are real constants V k. We can show the following proposition. 7. s X i , . . s having a uniform distribution on the interval [0,1] and Z := X -\-Y, then Sz = [0,2] and /•! oo / fx (u) fy {z-u)du= Since -oo / fy {z - u) du ^0 = fy {z-u) lifz — l

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Control Theory and Optimization I by M.I. Zelikin, S.A. Vakhrameev


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