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Problem: Find all solutions of the equation: \[ \sin z=\frac{4}{3} i \text {. } \]

10.3.12 Operations with complex numbers

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Problem: Expand \( f(z) \) into Laurant series (Taylor) in a given ring or neighborhood of a given point (in the latter case, indicate the area of convergence of the resulting series). \[ f(z)=\frac{z-4}{z^{2}(z+4)}, \quad 1<|z-1|<5 . \]

10.8.6 Series with complex terms

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Problem: Expand \( f(z) \) into the Laurant series (Taylor) in the given ring or the neighbourhood of the given point (in this case indicate the domain of convergence of the obtained series). \[ f(z)=\frac{z}{z+1}+e \frac{z}{z-1}, \quad z_{0}=1 . \]

10.8.7 Series with complex terms

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Problem: Expand \( f(z) \) into Laurent series (Taylor) into the given ring or neighborhood of the given point (in this case indicate the domain of convergence of the obtained series). \[ f(z)=\frac{1}{(z+1)(z-3)^{2}}, \quad 1<|z|<3 . \]

10.8.8 Series with complex terms

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Problem: Expand \( f(z) \) into Laurent series (Taylor) into the given ring or neighborhood of the given point (in this case indicate the domain of convergence of the obtained series). \[ f(z)=\frac{\cos z}{z^{2}}+\sin \frac{1}{z}, \quad z_{0}=0 . \]

10.8.9 Series with complex terms

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Problem: The system of random variables \( (X, Y) \) obeys the normal law with numerical characteristics \( M[X]=M[Y]=0, E_{x}=E_{y}=10, k_{x y}=0 \). Determine the probability that a) \( X0 ; Y<0 \).

15.5.21 Two-dimensional random variables and their characteristics

3.75 $

Problem: The probability of sensor failure within a month is \( 0,15.2000 \) sensors are inspected once in a month. How many spare sensors must be there, so that all sensors, having defects, could be replaced with the probability of at least 0,98 .

15.7.4 Limit theorems

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Problem: The distribution function of the random variable is equal to 0 when \( x<-1, x^{2}+2 x+1 \) when \( -1 \leq x<0 \), and 1 for other \( x \). Find the expected value of this random variable.

15.2.47 One dimensional random variables and their characteristics

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Problem: The distribution density of the random variable \( X \) is given. \[ f(x)=\left\{\begin{array}{cl} 0, & \text { when } x<0 \\ \alpha\left(x-\frac{x^{2}}{3}\right), & \text { when } 0 \leq x \leq 3 \\ 0, & \text { when } x>3 \end{array}\right. \] Find a) the coefficient \( \alpha \), b) the distribution function \( F(x) \), c) the expected value \( E(X) \) and the dispersion \( D(X) \).

15.2.48 One dimensional random variables and their characteristics

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Problem: The random variables \( X_{1} \) and \( X_{2} \) have binomial and Poisson distributions, respectively. Find the probabilities of \( p\left(3 \leq X_{i} \leq 5\right) \), if the expected value is \( M\left(X_{i}\right)=3 \), and the dispersion is \( D\left(X_{1}\right)=1,875 \).

15.2.49 One dimensional random variables and their characteristics

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Problem: The random variables \( X_{3} \) and \( X_{4} \) have an equal and normal distributions respectively. Find the probabilities of \( p\left(3

15.2.50 One dimensional random variables and their characteristics

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Problem: The continuous random variable \( X \) has the distribution function \( F(x)=0 \) when \( \quad x<1, \quad F(x)=1 \quad \) when \( x>4 \), \( F(x)=c_{1}+(x / 4)+c_{2}(x-1)^{2} \) when \( 1

15.2.51 One dimensional random variables and their characteristics

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Problem: Let the random variables \( X \) and \( Y \) be independent and given in accordance with the laws of distribution: \begin{tabular}{|c|c|c|c|} \hline\( x_{i} \) & 2 & 3 & 4 \\ \hline\( p_{i} \) & 0,3 & 0,2 & 0,5 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|c|c|} \hline\( y_{i} \) & 0 & 1 & 2 & 3 \\ \hline\( q_{i} \) & 0,2 & 0,4 & 0,1 & 0,3 \\ \hline \end{tabular} Find the distribution law of the variable \( Z=X-Y \) and the expected value and dispersion of the variables \( X, Y \) and \( Z \).

15.2.52 One dimensional random variables and their characteristics

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Problem: The deck of 36 cards is divided into two parts of 18 cards. In each part there turned to be two queens. Two cards were transferred from the \( 1^{\text {st }} \) to the \( 2^{\text {nd }} \) parts. Then the young rabbit took a card from the \( 2^{\text {nd }} \) half. Find the probability that it has taken a queen.

15.6.12 Definition and properties of probability

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Problem: There are three identical boxes. There are 2 white and 2 black balls in the \( 1^{\text {st }} \) box; there are 3 black balls in the \( 2^{\text {nd }} \) box and 1 black and 5 white balls in the \( 3^{\text {rd }} \) box. Someone, randomly choosing a box, takes out a random ball. What is the probability that the ball will be white?

15.6.13 Definition and properties of probability

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