Theory of random processes

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Problem: Find one-dimensional density, expected value and the dispersion of a random harmonic oscillation \( X(t)=\alpha \cos (\omega t+\phi) \), where \( \alpha, \omega \) are constant, \( \phi \) is a random phase, distributed evenly on \( [-\pi ; \pi] \).

15.1.31 Theory of random processes

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Problem: The correlation function \( K_{X}(\tau) \) of the stationary random function \( X(t) \) is given: \( K_{X}(\tau)=\sigma^{2} e^{-\alpha^{2} \tau^{2}} \). Find the correlation function of the random function \( Y(t)=a X^{\prime}(t) \).

15.1.32 Theory of random processes

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Problem: The random function \( X(t)=t+X_{1} \cos t+ \) \( +X_{2} \sin t \) is given, where the random vector \( \left(\mathrm{X}_{1}, \mathrm{X}_{2}\right) \) has the expected value \( (-1 / 2 ; 1) \) and the correlation matrix \( \left(\begin{array}{cc}3 & -2 \\ -2 & 2,9\end{array}\right) \). Construct a canonical expansion of the process, find its expected value, dispersion and correlation function:

15.1.33 Theory of random processes

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Problem: The spectral density of the stationary random function is given in the form: \( S_{X}(\omega)=\frac{D X}{\pi\left(\alpha^{2}+\omega^{2}\right)} \), where \( D X- \) dispersion of \( X \). Find the correlation function \( K_{X}(\tau) \).

15.1.34 Theory of random processes

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Problem: Find the expected value \( m_{Y}(t) \) and the spectral density \( S_{Y}(\omega) \) of the solutions of the equation \( Y^{\prime \prime}(t)+2 Y^{\prime}(t)+2 Y(t)=X^{\prime}(t)+X(t), \quad \) where \( X(t) \) is the stationary function, and \( m_{X}(t)=1 \), \( K_{X}(\tau)=e^{-|\tau|}(1+|\tau|) \).

15.1.35 Theory of random processes

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