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Problem: A random process \( Y(t)=a+X t \) is given, where \( X \) is a random variable: \( f(x) \sim N[2 ; 2] \). Find: \( M[Y(t)], K_{Y}\left(t_{1}, t_{2}\right), D[Y(t)] \), draw a family of trajectories of \( Y(t)=a+X t \).

15.1.10 Theory of random processes

3.05 $

Problem: A random process \( Y(t)=a t+X \) is given, where \( X \) is a random variable: \( f(x) \sim R[-2 ; 2] \). Find: \( \quad M[Y(t)], K_{Y}\left(t_{1}, t_{2}\right), D[Y(t)] \), draw the family of trajectories of \( Y(t)=a+X t \).

15.1.11 Theory of random processes

3.81 $

Problem: The random process \( X(t), t \geq 0 \), is defined by the formula \( X(t)=\alpha \cos (t+\beta)+\varepsilon \), where \( \alpha, \beta, \varepsilon- \) are independent random variables, moreover, \( \alpha \sim N(0,1), \varepsilon \sim N\left(0, \sigma^{2}\right), \beta \sim U[-\pi, \pi] \). Find: \( P\left(X\left(t_{1}\right) \leq X\left(t_{2}\right) \mid \alpha \geq 0\right) \), where \( 0 \leq t_{1} \leq t_{1} \leq \frac{\pi}{2} \).

15.1.12 Theory of random processes

3.81 $

Problem: The random process \( X(t), t \geq 0 \), is defined by the formula \( X(t)=\alpha \cos (t+\beta)+\varepsilon \), where \( \alpha, \beta, \varepsilon \) are independent random variables, moreover \( \alpha \sim N(0,1), \varepsilon \sim N\left(0, \sigma^{2}\right), \beta \sim U[-\pi, \pi] \). Is the process \( X(t), t \geq 0 \) stationary in broad sense?

15.1.13 Theory of random processes

6.36 $

Problem: A random process \( \xi(t)= \) const, \( n-1 \leq t \leq n \), \( \forall n \in \mathbb{N} \) is given. The values of \( \xi(t) \) when \( t \in(n, n+1] \) and \( t \in(m, m+1] \) are independent random variables \( (n \neq m) \), with a probability density \[ P(x)=\frac{|x|^{\lambda}}{2 \Gamma(x+1)} e^{-|x|} . \]

15.1.14 Theory of random processes

6.36 $

Problem: Find the expected value, correlation function and the variance of the random function \( X(t)=X_{1} \cdot e^{2 t}-X_{2} \cdot \cos 5 t+3 t^{2}-1 \), where \( X_{1} \) and \( X_{2} \) are uncorrelated random variables with characteristics: \( m_{X_{1}}=0,2, m_{X_{2}}=0,3, D_{X_{1}}=0,01 \), \( D_{X_{1}}=0,04 \).

15.1.15 Theory of random processes

3.81 $

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

5.09 $

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

6.36 $

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

3.05 $

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

6.36 $

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

5.09 $

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

6.36 $

Problem: The improper motion is given by \[ X^{\prime}=\frac{1}{3}\left(\begin{array}{ccc} 2 & 2 & -1 \\ 2 & -1 & 2 \\ -1 & 2 & 2 \end{array}\right) X+\left(\begin{array}{l} 4 \\ 0 \\ 2 \end{array}\right) \] It is a composition of symmetry with respect to the plane \( 2 x_{1}-4 x_{2}+a x_{3}=6 \) and parallel translation, moreover \( a=\cdots \).

5.1.4 Affine transformations

3.81 $

Problem: To obtain the canonical form of the hypersurface given by the equation \( 2 x_{1} x_{2}+x_{2}-x_{3}=0 \), it's necessary to move the origin of the coordinates to the point \( (a, b, c) \), where \( 2(a+b+c)=\cdots \).

5.1.5 Affine transformations

2.54 $

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