For the stochastic process {X(t)} of Example 4.2 with X(t) = A cos(ω t) in which A is a random variable, find μX (t), φXX (t,s), and φXX (t,s). Confirm that differentiating the moment functions for {X (t)} and analyzing the time histories of the derivative process {X (t)} give the same results.

Question

For the stochastic process [latex] \{X(t)\}[/latex] of Example 4.2 with [latex] X(t) = A cos(\omega t)[/latex] in which A is a random variable, find [latex] \mu _{ \dot{X} } (t)[/latex] , [latex] \phi_{X \dot{X} } (t,s)[/latex] , and [latex] \phi_{ \dot{X} \dot{X} } (t,s)[/latex]. Confirm that differentiating the moment functions for [latex] \{X (t)\}[/latex] and analyzing the time histories of the derivative process [latex] \{ \dot{X} (t)\}[/latex] give the same results.

Accepted Answer

From Example 4.2, we know that [latex] \phi_{XX} (t,s) = E(A^{2}) \cos (\omega t) \cos (\omega s)[/latex] and [latex]\mu _{X}(t) = \mu_{A} \cos (\omega t) [/latex] . Taking derivatives of these functions gives

[latex] \mu _{\dot{X} }(t)=-\mu _{A}\omega \sin (\omega t) [/latex], [latex] \phi _{X \dot{X} }(t,s)=-E(A^{2})\omega \cos (\omega t) \sin (\omega s) [/latex]

and

[latex] \phi _{\dot{X} \dot{X}}(t,s)=E(A^{2})\omega ^{2} \sin (\omega t) \sin (\omega s) [/latex]

The relationship for the time histories of [latex] \{\dot{X} (t)\}[/latex] is

[latex] \dot{X}(t) = -A \omega \sin (\omega t)[/latex]

and it is obvious that this relationship gives the same moment functions for [latex] \{ \dot{X}(t)\}[/latex] as were already obtained by differentiating the moment functions for [latex] \{X(t)\}[/latex].

Question 4.20: For the stochastic process {X(t)} of Example 4.2 with X(t) =...

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