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Problem: It's known that among 1 million adult residents of the city \( N \) approximately 2 thousand have the surname Kirby. By the method of random sampling a contingent of 1,000 people is formed for the jury. What is the probability that among the potential jury there will be at least three people with this surname?

15.2.42 One dimensional random variables and their characteristics

2 $

Problem: The probability density of the random variable \( X \) has the form: \[ f_{X}(x)=\left\{\begin{array}{lc} (x+a)^{2}, & \text { if }-a \leq x<0 \\ a^{2}-x^{2}, & \text { if } 0 \leq x \leq a \\ 0, & \text { otherwise } \end{array}\right. \] Find: a) the coefficient \( a \); b) probability \( P\{X \in(-0,5 ; 0,75)\} ; \) c) the expected value of \( \mathrm{M}[X] \) d) the distribution function \( f_{X}(x) \).

15.2.43 One dimensional random variables and their characteristics

3.75 $

Problem: It's known that \( X \sim N(1,1 / 2) \quad \) and \( P\{1-l \leq X \leq 1+l\}=0,9544 \). Find \( l \).

15.2.44 One dimensional random variables and their characteristics

1 $

Problem: It's known that a normally distributed random variable takes values, smaller than 248 , with the probability of 0,975 , and values, greater than 279 , with the probability of 0,005 . Find the probability density of this random variable.

15.2.45 One dimensional random variables and their characteristics

1.75 $

Problem: Calculate the expected value and the dispersion of the hypergeometric distribution.

15.2.46 One dimensional random variables and their characteristics

3.75 $

Problem: From an urn, containing 6 white and 4 black balls, Jon and Peter take out 3 balls, without returning them, in the following order: Peter - Jon - Jon. Random variables: \( X \) is the number of white balls, that Jon has, \( Y \) is the number of white balls, that Peter has. Describe the distribution law of the random vector \( (X, Y) \). Find the correlation coefficient \( p[X, Y] \).

15.5.20 Two-dimensional random variables and their characteristics

3 $

Problem: \( 30 \% \) of the patients at the hospital have disease \( K \), \( 20 \% \) of the patients have disease \( L \), and the rest of the patients have disease \( M \). The probabilities that the patients with diseases \( K, L, M \) will be discharged healthy, are equal to respectively 0,5 ; 0,\( 4 ; 0,9 \). Find the probability that a healthy patient who was discharged had disease \( K \).

15.6.6 Definition and properties of probability

1 $

Problem: There are black, white and blue balls in three boxes. It's known that in the \( 1^{\text {st }} \) box there are 2 white, 4 blue and 5 black balls, in the \( 2^{\text {nd }} \) box there are 4 white, 3 blue and 5 black balls, and in the \( 3^{\text {rd }} \) box there are 1 black, 2 white and 2 blue balls. From a randomly chosen box 3 balls are taken out. Find the probability that they are of the same colour.

15.6.7 Definition and properties of probability

1.25 $

Problem: There are white and black balls in two boxes. There are 2 black and 1 white balls in the \( 1^{\text {st }} \) box, and 3 white and 3 black balls in the \( 2^{\text {nd }} \) box. Two balls are randomly chosen from each of the two boxes and put them in the \( 3^{\text {rd }} \) empty box. It turns out that there are balls of both colours in the \( 3^{\text {rd }} \) box. Find the probability that a ball, randomly chosen from the \( 3^{\text {rd }} \) box, will be white.

15.6.8 Definition and properties of probability

3 $

Problem: There are three batches with the same number of elements. One batch contains second-rate elements, and the other two contain only first-rate elements. A randomly chosen element turned out to be first-rate. Find the probability that it is from the batch, containing second-rate elements.

15.6.10 Definition and properties of probability

2 $

Problem: Let \( \xi_{1}, \xi_{2}, \ldots \) be a sequence of independent random variables, each of which has the Poisson distribution. Prove that the series \( \sum_{n=1}^{\infty} \xi_{n} \) almost surely converges only when the series \( \sum_{n=1}^{\infty} E \xi_{n} \) converges.

15.7.1 Limit theorems

4.99 $

Problem: Let \( \xi_{1}, \xi_{2}, \ldots \) is the sequence of such independent non-negative random variables that \( p\left(\xi_{n}>c\right)=0 \). Prove that the series \( \sum \xi_{n} \) almost surely converges only when the series \( \sum E \xi_{n} \) converges.

15.7.2 Limit theorems

4.99 $

Problem: Find all values of the parameter \( a \), for each one of which the equation \( \left|\xi^{2}-49\right|-12|\xi-a|- \) \( -12 a=0 \) with respect to the variable \( \xi \) has exactly 4 solutions. If the question of the task allows several variants of answer, show them in the form of a set.

17.31 USE problems

3.75 $

Problem: Find all values of the parameter \( \omega \), for each one of which the equation \( \left|\frac{1}{2 \sigma}-3\right|-\omega \sigma-\frac{3}{2}=0 \) with respect to the positive value of \( \sigma \) has no solution. If the question of the task allows some variants of answer, show them in the form of a set.

17.32 USE problems

4.99 $

Problem: Solve the inequality \[ \frac{2-\sqrt{\left(2+\log _{2} h\right)\left(-1+\log _{2} h\right)}}{\log _{2} h}+1 \geq 0 \] with respect to the variable \( h \). If the question of the task allows several variants of answer, show them in the form of a set.

17.33 USE problems

1.5 $

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