Berry, J.; Haller, L.; Sveshnikov, Aram Aruti︠u︡novich's Applied methods of the theory of random functions PDF

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

ISBN-10: 1483197603

ISBN-13: 9781483197609

ISBN-10: 1483222632

ISBN-13: 9781483222639

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Example text

If this condition is n o t satisfied, then, as mentioned a t t h e end of t h e preceding section the formulae for the correlation function remain unchanged; it is only necessary t o take into account t h e value of x(t) when calculating the mathematical expectation of the derivative and integral. To do this we return to t h e original formulae. 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.

18) has been taken outside the sign of the mathematical expectation since the increments Ax and A2 are not random. 22) that is, an expression depending only on r. Consequently, the derivative of a stationary random function is also a sta­ tionary random function and instead of (22) we can write Kv(T)=-^£t. 23) y We pass to the consideration of the integral of a random function. 24) a t As was shown above, in consequence of the differentiability of X(t) the derivative Q7~Q7 exists for tx = t2: however, i t c a n b e 2 .

Y(t) = / X(tj) dt,. 0 Determine the variance of Y(t) for £ = 20 sec, if Kx{x) = ^ e - a l T l ( l + a | r | ) , ^ = 10 cm 2 /sec 2 , a = 0-5 sec- 1 . Applying (35) we have t D[Y(t)] = 2A J (*-r)

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

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