Theory of random processes

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Problem: The random function \( X(t) \) has the characteristics \( m_{X_{t}}=\cos 3 t+t^{4}-2 t+4, K_{X}\left(t_{1}, t_{2}\right)=4 \cdot e^{6\left(t_{1}^{2}+t_{2}^{2}\right)} \). Find the expected value, the correlation function and the dispersion of the random function \( Y(t)=\sin 3 t \cdot X^{\prime}(t)+5 t^{2} X(t)+t^{3}+1 \).

15.1.16 Theory of random processes

5.09 $

Problem: Find the expected value, correlation function and the dispersion of the random process \( Z(t)=t^{2} \cdot X(t)++Y^{\prime}(t) \), if the random processes \( X(t) \) and \( Y(t) \) are uncorrelated, with the characteristics: \[ \begin{array}{l} m_{X}(t)=\sin 7 t, \quad m_{Y}(t)=4 t^{3}-5 t^{2}, \\ K_{Y}\left(t_{1}, t_{2}\right)=t_{1}^{3} \cdot t_{2}^{3} \cdot e^{2\left(t_{1}^{2}+t_{2}{ }^{2}\right)} . \end{array} \]

15.1.17 Theory of random processes

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Problem: The random function \( X(t) \) is given by the canonical expansion \( X(t)=U\left(t^{3}+e^{t}\right)+ \) \( +V \cos t+t^{3} \), where \( D_{U}=0,1, D_{V}=0,6 \). Find the characteristics \( m_{Y}(t), K_{Y}\left(t_{1}, t_{2}\right), D_{Y}(t) \), if \[ Y(t)=\int_{0}^{t} \tau^{3} \cdot X(\tau) d \tau+\sin t \cdot X(t)+t^{4}-2 \]

15.1.18 Theory of random processes

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Problem: Find the expected values, correlation function and the dispersion of the random process \( X(t)=X \cdot \cos 5 t+X \cdot e^{-t^{2}}-1 \) where \( X \) is a random variable with the characteristics \( m_{X}=4,5, D_{X}=0,2 \).

15.1.19 Theory of random processes

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Problem: The random function \( X(t) \) is given by the canonical expansion \( X(t)=U \cdot e^{-t}+V \sin t+4 W \), where \( D_{U}=1, D_{V}=2, D_{W}=0,7 \). Find the characteristics of the random function \( Y(t): m_{Y}(t), K_{Y}\left(t_{1}, t_{2}\right), D_{Y}(t) \), where \[ Y(t)=2 X(t)+3 \int_{0}^{t} \tau X(\tau) d \tau \text {. } \]

15.1.20 Theory of random processes

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Problem: The random function \( X(t) \) has the characteristics \( m_{X}(t)=t^{5}+t \sin 2 t+2, K_{X}\left(t_{1}, t_{2}\right)=4 e^{-2\left(t_{1}-t_{2}\right)^{2}} \). Find the expected value, correlation function and the dispersion of the random process \( Y(t)=\cos t \cdot X(t)+3 t \cdot X^{\prime}(t)+4 t \).

15.1.21 Theory of random processes

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Problem: The random function \( X(t) \) is given by a canonical expansion \( X(t)=4 t^{2}+2 t-1+X_{1} \). \( \cdot(t-\sin 3 t)+X_{2} \cdot\left(t^{2}+e^{t}\right), \quad \) where \( \quad D_{X_{1}}=0,3 \), \( D_{X_{2}}=0,2 \). Find the characteristics \( m_{Y}(t) \), \( K_{Y}\left(t_{1}, t_{2}\right), D_{Y}(t) \), if \[ Y(t)=6 \cdot \int_{0}^{t} \tau \cdot X(\tau) d \tau+2 e^{-t} \cdot X(t)+\sin 4 t+2 t . \]

15.1.22 Theory of random processes

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Problem: Find the spectral density of the random function \( \xi(t) \), if \( K_{\eta \eta}(\tau)=e^{-|\tau|}: \) \[ \xi^{\prime \prime}+2 \xi^{\prime}+2 \xi=\eta^{\prime}+\eta . \]

15.1.23 Theory of random processes

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Problem: The action of a stationary linear system is given as follows: \( \xi^{\prime \prime}+2 \xi^{\prime}+2 \xi=\eta^{\prime}+\eta \). The correlation function of the output is known: \( K_{\eta, \eta}(\tau)=\exp (-|\tau|), \tau \in(-\infty ;+\infty) \). Find the correlation function of the input.

15.1.24 Theory of random processes

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Problem: Find the spectral density of the random function \( \xi(t) \), if \( K_{\eta, \eta}(\tau)=\exp (-2|\tau|) \). \[ \xi^{\prime \prime}+\xi=\eta^{\prime}+2 \eta \text {. } \]

15.1.25 Theory of random processes

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Problem: \( \xi^{I V}+4 \xi=\eta^{\prime}+2 \eta \). The random function of the output is \( \eta(t) \). Find the spectral density of the input \( S_{\xi}(\omega) \), if \( K_{\eta, \eta}(\tau)=e^{-2|\tau|} \).

15.1.26 Theory of random processes

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Problem: A spectral density of the random function \( \xi(t) \) is given: \[ S_{\xi}(\omega)=\frac{1}{\pi\left(4+\omega^{4}\right)} . \] Find the correlation function of this random function.

15.1.27 Theory of random processes

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Problem: The random function has the form \( X(t)=V t^{2}(t>0) \), where \( V \) is a random variable, evenly distributed on \( [0 ; 3] \). Find the onedimensional distribution function and onedimensional density of this process.

15.1.28 Theory of random processes

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Problem: Let the random function \( X(t) \) have the characteristics \( \quad m_{x}(t)=1, K_{x}(t, s)=e^{-(t-s)^{2}} \). Find the characteristics of the random functions \[ \begin{array}{l} Y=1+t-x, Z=t^{2}+x^{\prime} \sin t, U=t+x^{\prime \prime}, \\ V=\int_{0}^{t}(1+t) X(t) d t . \end{array} \] Find out wheter the functions \( X, Y, Z, U, V \) are stationary.

15.1.29 Theory of random processes

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Problem: The spectral density of the stationary random function \( X(t) \) is given: \[ S_{X}(\omega)=\frac{1}{\alpha \sqrt{\pi}} \cdot e^{-\frac{\omega^{2}}{4 \alpha^{2}}} . \] Find the correlation function of \( X(t) \).

15.1.30 Theory of random processes

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