Theory of random processes

MathProblemsBank banner

MathProblemsBank banner

Math Problems and solutions

Problem list Free problems

Attention! If a subsection is selected, then the search will be performed in it!

Problem: An elementary random function has the form \( Y(t)=a X+t \), where \( X \) is a random variable, distributed in accordance with the normal law with the parameters \( m, \sigma\left(p(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-m)^{2}}{2 \sigma^{2}}}\right) \), \( a \) is a non-random value. Find the characteristics of the elementary random function \( Y(t) \).

15.1.1 Theory of random processes

3.81 $

Problem: Let \( S_{0}=0, S_{k}=\xi_{1}+\cdots+\xi_{k}, 1 \leq k \leq n \), where \( \xi_{1}, \ldots, \xi_{k} \) are independent normally distributed, \( \mathcal{N}(0,1) \), random variables. Let \( \phi(x)=P\left\{\xi_{1} \leq x\right\} \), \( \mathcal{F}_{k}=\sigma\left(\xi_{1}, \ldots, \xi_{k}\right), 1 \leq k \leq n, \mathcal{F}_{0}=\{\emptyset, \Omega\} \). Show that for any \( a \in R \) the sequence \( X=\left(X_{k}, \mathcal{F}_{k}\right)_{0 \leq k \leq n} \) with \( X_{k}=\phi\left(\frac{a-S_{k}}{\sqrt{n-k}}\right) \quad \) is a martingale.

15.1.2 Theory of random processes

6.36 $

Problem: A random process \( X(t)=u \cos t+v e^{t}+t \), is given, where \( u \) and \( v \) are random variables with \( M(u)=M(v)=2 ; \quad D(u)=D(v=0.2) \), \( \operatorname{cov}(u, v)=0,1 \). Find the characteristics of the random process \( Y(t)=2 X^{\prime}(t)-2 \).

15.1.3 Theory of random processes

3.81 $

Problem: The random process \( X(t) \) is given by the canonical expansion \( \quad X(t)=(t+1) u+\sin t \cdot v+e^{-3 t} \), where \( D(u)=D(v)=3 \). Find: a) the characteristics of the random process \( X(t) \); b) the characteristics of the random process \[ Y(t)=e^{2 t} \int_{0}^{t} X(\tau) d \tau \text {. } \]

15.1.4 Theory of random processes

5.09 $

Problem: Let \( X(t) \) be a stationary random function with a correlation function \( K_{x}(\tau)=e^{-\alpha|\tau|}, \alpha>0 \). Find the spectral density \( S_{x}(\omega) \).

15.1.5 Theory of random processes

3.05 $

Problem: The random function \( X(t) \) has the characteristics \( m_{X}(t)=0, \quad k_{x}\left(t, t^{\prime}\right)=\frac{1}{1+\left(t-t^{\prime}\right)^{2}} \). Find the characteristics of the random function \( Y(t)=\int_{0}^{t} X(t) d t \) and find out whether \( X(t), Y(t) \) are stationary?

15.1.6 Theory of random processes

3.81 $

Problem: The random process has the form: \( X(t)=V \cdot e^{-t}+a t^{2} \), where \( V \) is a random value, distributed according to the exponential law with the parameter \( \lambda, a= \) const. Find the expected value, the mathematical expectation, the normalized autocovariance function and the dispersion of \( t^{2} X(t) \).

15.1.7 Theory of random processes

3.81 $

Problem: The characteristics of the random process \( X(t) \) are known: \( m_{X}(t)=2 t^{2}-1, R_{X}\left(t_{1}, t_{2}\right)=2 e^{-3 t_{1}-t_{2}} \). Find the expected value and the dispersion of the process \( Y(t)=\frac{d X(t)}{d t}-2 \).

15.1.8 Theory of random processes

2.54 $

Problem: A random process \( Y(t)=a+X t^{2} \) is given, where \( X \) is a random variable: \( f(x) \sim R[0 ; 3] \). 1) Draw the trajectory of \( Y(t) \), 2) Find: \( M[Y(t)], K_{Y}\left(t_{1}, t_{2}\right), D[Y(t)] \).

15.1.9 Theory of random processes

3.81 $

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 $

1
2
3
›