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Problem: The batch contains 20 televisions, among which 6 have a defect. Two televisions have been bought. Compose the series of serviceable televisions among the bought ones. Find numerical characteristics of the given random variable.

15.2.1 One dimensional random variables and their characteristics

2 $

Problem: Independent random variables \( x \) and \( y \) have the distribution \begin{tabular}{|c|c|c|} \hline\( x_{i} \) & -2 & -1 \\ \hline\( p_{i} \) & 0,4 & 0,6 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|} \hline\( y_{i} \) & 1 & 2 \\ \hline\( p_{i} \) & 0,7 & 0,3 \\ \hline \end{tabular} Find the law of distribution of the random variable \( Z=-x+2 y \). Find \( F(z), P(3 \leq z \leq 4) \).

15.2.2 One dimensional random variables and their characteristics

1.25 $

Problem: The distribution density of the random variable \( \mathrm{X} \) is given: \[ f(x)=\left\{\begin{array}{cc} 0, & x \leq 2 \\ 2 x-4, & 23 \end{array}\right. \] Find \( F(x), M(x), D(x), \sigma(x) \).

15.2.3 One dimensional random variables and their characteristics

2 $

Problem: The transition matrix of the chain is given, plot the graph with the given probabilities and calculate \( P_{i}(k), i=1,2,3 ; k=1,2 \), assuming that initially the system is in the state of \( S_{1} \). \[ \left(P_{i j}\right)=\left(\begin{array}{cccc} 0.2 & 0.3 & 0.5 & 0 \\ 0.6 & 0 & 0 & 0.4 \\ 0.4 & 0.2 & 0.3 & 0.1 \\ 0.5 & 0 & 0 & 0.5 \end{array}\right) \text {. } \]

15.3.1 Markov chains

2.5 $

Problem: A labeled graph of system states is given. Find: a) the transition matrices in one and two steps, b) the probabilities of the system state after the first, second and the third steps, if at the initial moment the system was in the state of \( S_{2} \), c) final probabilities.

15.3.2 Markov chains

4.99 $

Problem: Comprise the system of Kolmogorov equations for the probabilities of states of the continuous Markov chain, using the graph of states; find these probabilities, if at the initial moment the system was in the state of \( S_{2} \).

15.3.3 Markov chains

4.99 $

Problem: A labeled graph of a system states is given. Find: a) the transition matrix in one and two steps, b) the probabilities of the system state after the first, second, third steps, if at the initial moment the system was in the state of \( S_{1} \), c) final probabilities.

15.3.4 Markov chains

4.99 $

Problem: For the given propositional logic formulas, build the corresponding logic functions in the form of truth tables, determine the validity, satisfiability (non-satisfiability) and the number of models of the formula: \[ f(p, q)=p \&(q \vee \neg p) \&((\neg q \rightarrow p) \rightarrow q) . \]

6.1.1.1 Propositional calculus

1.5 $

Problem: Write the following statements in the form of propositional logic formulas, build a truth table and determine the validity, satisfiability (nonsatisfiability) and the number of models of the obtained formulas: "If the workers or the administration persist, the strike will be settled when and only when the government gets an injunction, but no troops are sent to the factory."

6.1.1.2 Propositional calculus

2 $

Problem: Prove the non-satisfiability (or satisfiability) of the following sets of clauses by using the resolution method. Apply an arbitrary order of enumeration of clauses, as well as, at the instruction of the teacher, one of the following strategies: preference for monomials, linear, level saturation. \[ \{(q \vee \neg \tau),(\neg q \vee \neg \tau),(q \vee \tau),(\neg p \vee \neg \tau), \neg q\} . \]

6.1.1.3 Propositional calculus

1.5 $

Problem: Write down formally the following reasoning in the language of propositional logic and prove its validity using the method of resolutions. Premise: wages will increase only if there is inflation. If there is inflation, the cost of living will increase. Conclusion: The cost of living will increase.

6.1.1.4 Propositional calculus

2.5 $

Problem: The following notation is an expression (formula) of the propositional algebra: 1) \( ((A \wedge B) \Rightarrow C \) 2) \( A \Rightarrow \wedge B \Leftrightarrow C \); 3) \( A-B \wedge \bar{A} \); 4) \( A \vee(B \Rightarrow \bar{A}) \).

6.1.1.5 Propositional calculus

2.5 $

Problem: Find curvature of the line \( \rho=a(1+\cos \varphi) \).

7.1 Differential geometry

2.5 $

Problem: Compose an equation for the evolutes of the line. \[ \rho=a(1+\cos \varphi) \text {. } \]

7.2 Differential geometry

3.75 $

Problem: Compose a natural equation of a curve given parametrically: \[ x=a(\cos t+t \sin t), y=a(\sin t-t \cos t) . \]

7.3 Differential geometry

2.5 $

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