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Problem: A «wrong» coin (the probability of getting "heads" is \( A \) ) is tossed for \( N \) times. The following variables are considered: \( x \) is the number of "heads", \( y \) is the number of "tails", \[ z_{1}=\frac{x}{y}, z_{2}=x+y, z_{3}=\frac{x}{z_{2}} \text {. } \] Answer the following questions about these random variables: a) describe the distributions of the random variables \( x, y, z_{1}, z_{2}, z_{3} \); find the expected values, the second moments, dispersions; b) describe the conditional distribution of the random variables \( x \mid y \); c) as a result of the \( M \)-th toss, it turned out that there are exactly \( L \) «heads», what is the probability that there will be no more than \( K \) "tails"? d) Let's find the covariance and the correlation coefficient of the variables \( x \) and \( y \); e) Find the covariance and the correlation coefficient of the variables \( x^{2} \) and \( y \); \[ A=0,69 ; N=252 ; M=142 ; L=80 ; K=55 \text {. } \]

15.5.2 Two-dimensional random variables and their characteristics

12.85 $

Problem: A two-dimensional random variable \( (\xi, \eta) \) is uniformly distributed in the triangle \( A B C \) with the vertices \( A(-1,0), B(0,2), C(0,-2) \). Find the distribution function of the probability \( (\xi, \eta) \), the marginal densities of the distribution \( \xi, \eta \), the expected values, dispersions, covariance and the correlation coefficient. Are the variables \( \xi, \eta \) independent?

15.5.3 Two-dimensional random variables and their characteristics

5.91 $

Problem: Three passengers get on the train, randomly choosing one of the six carriages. What is the probability that at least one of them will get on the first carriage, if it's known that they have got into different carriages?

15.6.1 Definition and properties of probability

1.03 $

Problem: It's known that a 5-digit telephone number has different digits. Under this condition what is the probability that exactly one of the numbers is even ( 0 is considered an even number and the telephone number can begin with a 0 ).

15.6.2 Definition and properties of probability

1.03 $

Problem: The central accounting department of the corporation received packs of receipts for verification and processing. \( 90 \% \) of the packs were found to be satisfactory: they contained \( 1 \% \) of incorrectly formalized receipts. The rest \( 10 \% \) of the receipts were found to be dissatisfying, i.e., they contained \( 5 \% \) incorrectly formalized receipts. What is the probability that a randomly chosen receipt will be incorrectly formalized?

15.6.3 Definition and properties of probability

1.03 $

Problem: There are 5 black and 6 white balls in the box. 4 balls are taken out. Find the probability that there are at least two white balls among those, taken out.

15.6.4 Definition and properties of probability

1.28 $

Problem: The lifespan of an electric lamp has an exponential distribution with a mathematical expectation of \( L \) hours. What is the possibility that the lamp will last from \( m_{1} \) to \( M_{1} \) hours if \[ L=76 ; m_{1}=75 ; M_{1}=109 \text {. } \]

15.2.4 One dimensional random variables and their characteristics

1.03 $

Problem: An experiment is performed in which a random occurrence \( A \) can take place with \( p \) probability. The experiment is repeated under the same conditions \( n \) times. \( n=1000 ; p=0,6 \). Determine the probability that the occurrence \( A \) will take place at least 580 times.

15.2.5 One dimensional random variables and their characteristics

1.28 $

Problem: The random variable \( X \) in the interval \( (0 ; 1) \) is given by the distribution density \( f(x)=2 x \), outside this interval \( f(x)=0 \). Find the initial and central moments of the first, second, third and fourth orders.

15.2.6 One dimensional random variables and their characteristics

2.57 $

Problem: Will the operator \( (A x)(t)=\int_{0}^{1} \sin (t x(s)) d s \) contracting in the space \( C([0 ; 1]) \) with respect to the uniform metric?

19.3.1 Linear operators

3.08 $

Problem: Find all the values of the parameter \( \alpha>0 \), for which the operator \( T: L^{2}[0 ; 1] \rightarrow L^{2}[0 ; 1] \), \( (T f)(x)=x f(x)-f\left(x^{\alpha}\right) \) is continuous.

19.3.2 Linear operators

3.85 $

Problem: Find the spectrum of the operator \( A: l_{2} \rightarrow l_{2} \), \( A\left(x_{1}, x_{2}, \ldots, x_{n}, \ldots\right)=\left(x_{1}, 0, x_{2}, 0, \ldots, x_{n}, 0, \ldots\right) \).

19.3.3 Linear operators

3.08 $

Problem: Prove the linearity and calculate the norm of the functional \( f(x)=\int_{0}^{2} t^{2} x(t) d t \), where a) \( x \in L_{1}[0,2] \), b) \( x \in L_{2}[0,2] \), c) \( x \in C[0,2] \).

19.3.4 Linear operators

6.42 $

Problem: Prove the linearity and calculate the norm of the operator \( (A x)(t)=x(t) \), where a) \( A: C^{(1)}[0 ; 1] \rightarrow C[0,1] \), b) \( A: C[0,1] \rightarrow L_{1}[0,1] \).

19.3.5 Linear operators

3.34 $

Problem: Prove that the sequence of operators \[ A_{n} x=(\underbrace{0, \ldots, 0}_{n}, x_{1}, \ldots, x_{n}, 0,0, \ldots) \] where \( A_{n}: l_{1} \rightarrow l_{1} \), doesn't converge pointwise to the zero operator.

19.3.6 Linear operators

2.57 $

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