Applied methods of the theory of random functions, Edition: by Berry, J.; Haller, L.; Sveshnikov, Aram Aruti︠u︡novich

By Berry, J.; Haller, L.; Sveshnikov, Aram Aruti︠u︡novich

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Extra resources for Applied methods of the theory of random functions, Edition: [1st English ed.]

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39) Thus a general rule can be formulated: the mathematical expectation of the derivative (integral) of a random function is equal to the derivative (integral) of the mathematical expectation of this function. All the results of this section were obtained without any assumptions having been made about the character of the density distributions of the ordinates of the random functions. Accordingly they are applicable both to normal and to nonnormal random functions. However, normal random func­ tions possess features which considerably simplify the use of these relations.

I n t h e case when t h e differential equation has constant coefficients t h e particular integrals of t h e homogeneous equation are of the form eXjt (we consider for simplicity t h a t all the roots of the characteristic equation are different — t h e occurrence of multiple roots does n o t change t h e final re­ sult) a n d direct substitution in (21) shows t h a t p(t, t±) is a linear combination (withconstant coefficients) of expressions of the form e^-**), t h a t is, it is a function of the difference t-tv Hence in this case t Vi(t) = $p{t-h)x(h)dti to t-t0 = j p(t)x(t-T)dt.

0 This formula shows that the variance of the angle of list does not depend on t; this seems reasonable in view of the assump­ tion of the steadiness of rolling. 3. t T(t) = a^W + ^ ^ ^ + bAe-^XitJd^ + C. o Determine the correlation function Ky(tly t2) if the correla­ tion function Kx{tlf t2) is known. Since T(t) is obtained from X(t) by the application of the linear (non-homogeneous) operator t L1== _d_ 6 a o + % - dt ^r+ i e Wl * i + C = L + <7 we have from the general formula (34) Expanding the product of operators dt + a0bA \e-u'dt'+ I e~u" V0 ■+ dt"\ + 0 we obtain finally ■*M*l> h) = «o A x(n» h)+aoai\ g»Jgxfa, t2) ##!

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