А) индексы структуры

Date: 2015-10-07; view: 351.

3,9

167,29

10,95

0,5

0,9

3,3

0,25

Statistics

Вопрос 12

1. A random variable Х is distributed under a normal law. Find by method of maximal plausibility the pointwise estimator of unknown parameter of a normal distribution on the following sample:

1. A random variable Х is distributed under a normal law. Find by method of maximal plausibility the pointwise estimator of unknown parameter of a normal distribution on the following sample:

Ds = 7,21

1. A random variable Х is distributed under Poisson law. Find by method of moments the pointwise estimator of unknown parameter of Poisson distribution on the following sample:

1. A random variable Х is distributed under Poisson law. Find by method of moments the pointwise estimator of unknown parameter of Poisson distribution on the following sample:

1. A reliable event is: an event that necessarily will happen if a certain set of conditions S holds.
2. A sample is extracted from a parent population:

Find an unbiased estimator of parent population (the sample mean ). 4

1. A sample is given as a discrete variation series:

Find the size of the sample: 25

1. A sample is given as a distribution of frequencies:

Find the distribution of relative frequencies.

1. A sample is given as a distribution of frequencies:

Find the sample mean. 5,9

1. A sample is given as a distribution of frequencies:

Find the sample dispersion. 2,49

1. A sample is given as distribution of frequencies:

Find the distribution of relative frequencies.

1. A sample is given by the following table:Find an unbiased estimator of dispersion s² (the revised dispersion).

1. A sample is given by the following table:Find the sample dispersion D_s.

42. A sample is given by the following table:

Find the sample mean = 5,76

1. A student knows 8 of 11 questions of an exam. Find the probability that he knows 3 randomly chosen questions. 56/165

A*B s 4ertoi + *B

1. An estimator of parameter is called unbiased if: its mathematical expectation is equal to the estimated parameter, i.e.
2. An estimator of parameter is called biased if:
3. An estimator of parameter is called consistent if: it satisfies the law of large numbers, i.e. it converges on probability to the estimated parameter:
4. An impossible event is: event that certainly will not happen if the set of conditions S holds.
5. An improper integral of density of distribution in limits from – ¥ till ¥ is equal to integral ot minus besk do plus besk φ (x)dx=1
6. An unbiased estimator of dispersion found on a sample of size n = 41 is equal to Find a biased estimator of dispersion D_s. 2,9268
7. An unbiased estimator of dispersion found on a sample of size n = 41 is equal to Find a biased estimator of dispersion D_s. 7,81
8. At testing statistical hypotheses a mistake of the first sort consists of: the correct null hypothesis will be rejected
9. At testing statistical hypotheses a mistake of the second sort consists of: the incorrect null hypothesis will be accepted.
10. At testing statistical hypotheses the hypothesis that contradicts the null hypothesis is called: competing (alternative).
11. Calculate : 455
12. Calculate : 30

D(X) = integral ot minus besk do plus besk (x-a)^2 φ(x)

1. Dispersion D of a variation series is: the arithmetic mean of squares of deviations of variants from their arithmetic mean
2. Dispersion of a constant C is 0
3. Dispersion of a continuous random variable X of which possible values belong to the entire axis OX is defined by the equality
4. Dispersion of a discrete random variable X is M(X^2)-[M(X)]^2 or M(X-M(X))^2
5. Distribution of a discrete random variable X is given by the table:

Find the mathematical expectation M(X). -0,5

1. Distribution of a discrete random variable X is given by the table:

Find the dispersion D(X). 0,25

1. Distribution of a discrete random variable X is given. What is M(X) equal to?

	0.4	0.3	0.3

1. Electric lamps are made at two factories, and the first of them delivers 70%, and the second - 30% of all consumed production. 83 of each hundred lamps of the first factory are standard on the average, and 63 - of the second factory. Find the probability that a bought lamp will be standard. 0,77

F(X) = integral ot minus besk do x f(t)dt

1. Find the confidence interval with reliability = 0,95 for estimating mathematical expectation of a normally distributed attribute X if parent mean square deviation s = 5, sample mean , sample size n = 25. By Table t = 1,96 has been found. 12,04<a<15,96
2. Find the confidence interval with reliability = 0,95 for estimating mathematical expectation of a normally distributed attribute X if the parent mean square deviation s = 10, sample mean , sample size n = 144. By Table t = 1,96 has been found. 108,37<a<111,63
3. Find the confidence interval with reliability = 0,95 for estimating mathematical expectation of a normally distributed attribute X if the parent sample mean square deviation s = 9, sample mean = 18,31, sample size n = 49. By Table t = 1,96 has been found. 15,79<a<20,83
4. Find the confidence interval with reliability = 0,95 for estimating mathematical expectation of a normally distributed attribute X if parent mean square deviation s = 3, sample mean = 4,1, sample size n = 36. By Table t = 1,96 has been found. 3,12<a<5,08
5. Find the confidence interval with reliability = 0,99 for estimating mathematical expectation of a normally distributed attribute X if parent mean square deviation s = 4, sample mean , sample size n = 16. By Table has been found. 7,63<a<12,77
6. Find the confidence interval with reliability = 0,99 for estimating mathematical expectation of a normally distributed attribute X if the revised sample mean square deviation , sample mean , sample size n = 9. By Table 3,35 has been found. 23,4<a<36,8
7. Find the confidence interval with reliability = 0,93 for estimating mathematical expectation of a normally distributed attribute X if the parent mean square deviation s = 9, sample mean , sample size n = 100. By Table t = 1,81 has been found. 88,371<a<91,629
8. Find the confidence interval with reliability = 0,94 for estimating mathematical expectation of a normally distributed attribute X if the parent mean square deviation s = 11, sample mean , sample size n = 121. By Table t = 1,88 has been found. 108,12<a<111,88
9. Find the confidence interval with reliability = 0,95 for estimating mathematical expectation of a normally distributed attribute X if the parent sample mean square deviation s = 5, sample mean = 18,71, sample size n = 25. By Table t = 1,96 has been found. 16,75<a<20,67

73. Find the confidence interval with reliability = 0,95 for estimating mathematical expectation of a normally distributed attribute X if the parent sample mean square deviation s = 7, sample mean = 18,51, sample size n = 100. By Table t = 1,96 has been found. 17,138<a<19,882

1. Find the confidence interval with reliability = 0,95 for estimating mathematical expectation of a normally distributed attribute X if the revised sample mean square deviation s = 18, sample mean = 980, sample size n = 16. By Table 2,12 has been found. 970,46<a<989,54
2. Find the confidence interval with reliability = 0,99 for estimating mathematical expectation of a normally distributed attribute X if the parent mean square deviation s = 5, sample mean =16,8, sample size n = 25. By Table t = 2,57 has been found. 14,23<a<19,37

Find the empirical function :

1. Find the median of a sample: {0, 1, 3, 3, 3, 4, 6, 10, 10, 12, 12, 15}. 5
2. Find the median of a sample: {10, 10, 15, 16, 16, 16, 16}. 16
3. Find the median of a sample: {12, 12, 12, 24, 26, 34, 37, 38, 49, 100}. (26+34)/2= 30
4. Find the median of a sample: {2, 2, 2, 4, 6, 7, 8, 9, 10}. 6
5. Find the median of a sample: {5, 13, 17, 17, 17, 17, 20, 20, 20, 20}. 17
6. Find the mode of a sample: {10, 10, 15, 16, 16, 16, 16, 16}. 16
7. Find the mode of a sample: {2, 2, 3, 4, 4, 4, 4, 4, 5}. 4
8. Find the mode of a sample: {25, 25, 30, 42, 42, 42, 42, 42, 57}. 42
9. Find the mode of a sample: {5, 8, 10, 10, 10, 10, 12, 12, 17}. 10
10. Find the mode of the following variation series:

Mo = 7

86. For an event - landing two heads at tossing two coins - the opposite event is: at least one tail

1. Function of distribution of a random variable X is given by the formula Find density of distribution f(x). proizvodnuyu naiti 2cos2x
2. How is called the variant which has the greatest frequency? moda
3. How is called the variant which partitions a ranked series into two parts that are equal by numbers of variants? mediana
4. How is changed the arithmetic mean if all the variants are decreased in 6 times? Decrease in 6 times
5. How is changed the arithmetic mean if all the variants are increased in 5 times? Increase in 5 times
6. How is changed the arithmetic mean if we add 15 to all the variants? Increase by 15
7. How is changed the arithmetic mean if we subtract 10 from all the variants? Decrease by 10
8. How is changed the sample dispersion if all the variants are decreased in 6 times? decreased in 36 times
9. How is changed the sample dispersion if all the variants are increased in 6 times? increased in 36 times
10. How is changed the sample dispersion if we add 6 to all the variants? Nothing
11. How is changed the sample dispersion if we subtract 10 from all the variants? Nothing
12. How is changed the sample dispersion if we subtract 6 from all the variants? nothing
13. How many 5-place telephone numbers are there if the digit “0” is not used on the first place? 90000
14. How many different 6-place numbers are possible to compose of digits 1, 2, 3, 4, 5, 6 if digits are not repeated? 720
15. How many ways are there to choose three employees on three different positions from 10 applicants? 720
16. If A and B are independent events then for Р(АВ) one of the following equalities holds: P(A)*P(B)
17. If an estimator of parameter has the least dispersion among all possible estimators of the parameter q calculated on samples of the same size n then it is called: efficient
18. If an event A can happen only provided that one of pairwise incompatible events В₁, В₂, В₃ forming a complete group will occur, Р(А) is calculated by the following formula: kak v 153 voprose
19. If D_s is the sample dispersion then an unbiased and consistent estimator of a parent dispersion is the revised sample dispersion determined by the formula: Ds = S^2* (n-1)/n
20. If pairwise incompatible events form a complete group, the sum of their probabilities is equal to: 1
21. If the probability of a random event A is equal to P(A), the probability of the opposite event is equal to: 1-P( )
22. If x_i is the number of appearances of an event A in the i^th experience, m is the number of trials in one experience, n is the number of experiences then the pointwise estimator of parameter p (probability of appearance of the event A in one trial) of a binomial law is: p*=Zxi/nm

Integral ot a do b x φ(x)ds

1. It is known that 10% of all radio lamps are non-standard. Find the probability that there will be no more than 1 non-standard lamp among 4 randomly taken radio lamps. 0,6561
2. It is known that M (X) = – 2 and M (Y) = 4. Find M (2X – 3Y). -16
3. Let f(x) be a density of distribution of a continuous random variable X. Then the function of distribution is:
4. Let A and B be events connected with the same trial. Show the event that means simultaneous occurrence of A and В. P=AB
5. Let A and B be events connected with the same trial. Show the event that means occurrence of only one of events A and B.
6. Let A and B be opposite events. Find Р(В) if Р(А) = 1/6. 5/6
7. Let a discrete variation series be given:

Find the arithmetic mean. 0,4

1. Let a discrete variation series be given:

Find the dispersion. 0,6

1. Let a discrete variation series be given:

Find the dispersion. 1,81

1. Let a discrete variation series be given:

Find the mode of the variation series: 3

1. Let a discrete variation series be given:

Find the mode of the variation series: 2

1. Let a discrete variation series of a sample be given:

Find the median of the variation series: 1

1. Let a discrete variation series of a sample be given:

Find the median of the variation series. 2

1. Let a discrete variation series of a sample be given:

Find the mode of the variation series: 1

1. Let a discrete variation series of a sample be given:

Find the mode of the variation series: 1

1. Let a discrete variation series of a sample be given:

Find the sample mean of the variation series: 1,12

125.Let a discrete variation series of a sample be given:

Find the sample mean of the variation series: 2

1. Let a discrete variation series of a sample be given:

Find the sample dispersion of the variation series: 1

1. Let a discrete variation series of a sample be given:

Find the size of the variation series: 25

1. Let a distribution of an attribute Х – the number of transactions on the stock exchange for the quarter received by n = 400 observations be given:Find the arithmetic mean . = 1,535
2. Let a variation series of a sample be given:

Find sample dispersion D_s. 1,8

1. Let a variation series of a sample be given:Find the sample mean . = 4
2. Let a variation series of a sample be given:

Find the sample mean: 4

1. Let D(X) = 2. Then D(X – 2) is equal to 2
2. Let D(X) = 3. Then D(2X) is equal to 12
3. Let M(X) = 2. Find M(2X). 4
4. Let M(X) = 2. Find M(X – 2). 0
5. Let n be the number of all outcomes, m be the number of the outcomes favorable to the event A. The classical formula of probability of the event A has the following form: P(A) = m/n
6. Let random variables X and Y with Y = 3X – 1 and D(X) = 2 be given. Find D(Y).18
7. Let results of sample checks of one-day profit of urban shops be given: {5, 5, 10, 10, 10, 10, 25, 30}. Find the mode. 10
8. Let results of sample checks of one-day profit of urban shops be given: {5, 5, 10, 14, 18, 20, 25}. Find the median. 14
9. Let the following distribution of a sample be given:

X<=1 F*(x) = 0

1<x<=4 F*(x) = 1/5

4<x<=6 F*(x) = ½

x>6 F*(x) = 1

141.Let the following distribution of a sample be given:

Find the empirical function X<=4 = 0; 4<x<=7 = 5/10 = ½ ; 7<x<=8 F*(x) = 7/10; x>8 F*(x) = 1

1. Let А₁, А₂, А₃ be events connected with the same trial. Let A be the event that means occurrence only one of events А₁, А₂ and А₃. Express the event A by the events А₁, А₂ and А₃. 1* 2*A3 + 1*A2* 3+A1* 2* 3
2. Let А₁, А₂, А₃ be events connected with the same trial. Let A be the event that means none of events А₁, А₂ and А₃ have happened. Express the event A by the events А₁, А₂ and А₃ vse A s 4ertami
3. Mathematical expectation of a continuous random variable X of which possible values belong to the entire axis OX is determined by the equalityM(x) = integral ot minus besk do plus besk x φ(x)dx
4. Mathematical expectation of a normally distributed random variable X is 3, and mean square deviation is 2. Write the density of distribution X. f(x) = 1/2√2П*e^-(x-3)^2/8
5. Mean square deviation of a random variable X is determined by the following formula √Ds
6. One letter is randomly chosen from the word "COMEDY". What is the probability that this letter is "U"?0
7. Pairwise incompatible events A₁, A₂ and A₃ form a complete group, and P(A₁) = 0,3; P(A₃) = 0,4. Then P(A₂) is equal to: 0,3
8. Probabilities of opposite events A and satisfy the following condition: P(A)+P( )=1; P(A)-P( )=Ø; P(A)*P( )=0
9. Probability to fail exam for the first student is 0,3; for the second – 0,5; and for the third – 0,1. What is the probability that only one of them will pass the exam? 0,185
10. Show one of true properties of mathematical expectation (C is a constant) M( C ) = C
11. Show the Bernoulli formula
12. Show the formula of total probability:
13. Show the mathematical expectation of a discrete random variable X: M(X) = Zxipi
14. The arithmetic mean of a variation series is: the sum of products of all variants on the corresponding frequencies divided on the sum of frequencies:
15. The dispersion D of a variation series can be found by the following formula:
16. The dispersion D(X) of a random variable X is equal to 2,25. Find s (Х): 1,5
17. The law of distribution of a discrete random variable X is given, M (X) = 6. Find x₁.2

	x1
	0.3	p2	0.5

1. The law of distribution of a discrete random variable X is given.

	-1
	0.2	0.5	Y

Find Y. 0,3 summa P=1

1. The letters T, A, O, M have been written on four cards. The cards are shuffled and randomly put in a row. What is the probability that the word "ATOM" will be in the row? 1/24
2. The mathematical expectation of a continuous random variable X of which possible values belong to an interval [a, b] is
3. The probability of a random event A is the number: 0<p<1
4. The probability of a reliable event is the number: 1
5. The probability of an arbitrary event cannot be: less than 0, more than 1
6. The probability of delay for the train №1 is equal to 0,2, and for the train №2 - 0,64. Find the probability that at least one train will be late. 0,872
7. The probability of delay for the train №1 is equal to 0,2, and for the train №2 - 0,64. Find the probability that both trains will be late. 0,128
8. The probability of hit in 10 aces for a shooter at one shot is 0,8. Find the probability that for 10 independent shots the shooter will hit in 10 aces exactly 7 times. 0,201
9. The probability of impossible event is the number: 0
10. The random variable X is given by an integral function of distribution: Find the probability of hit of the random variable X in the interval (0; 1): 1/3
11. There are 10 white, 15 black, 10 yellow and 25 red balls in a box. Find the probability that a randomly taken ball is black. ¼
12. There are 10 white, 15 black, 20 blue and 25 red balls in an urn. One ball is randomly extracted from the urn. Find the probability that the extracted ball is white or black. 5/14
13. There are 1000 tickets in a lottery. 500 of them are winning, and the rest 500 are non-winning. Two tickets have been bought. What is the probability that both tickets are winning? 0,2498
14. There are 12 white, 10 black, 10 yellow and 20 red balls in a box. Find the probability that a randomly taken ball is white. 3/13
15. There are 15 details in a box, and 10 of them are painted. Three details are randomly extracted from the box. Find the probability that the extracted details are painted. 24/91
16. There are 3 defective lamps among 10 electric lamps. Find the probability that two randomly chosen lamps will be defective. 1/15
17. There are 4 books on mathematics and 6 books on chemistry on a book shelf. Three books are randomly taken from the shelf. Find the probability that all taken 3 books are on mathematics. 1/30
18. There are 4 standard and 3 non-standard details in a box. Two details are randomly taken from the box. Find the probability that only one detail is standard. 4/7
19. There are 4 white and 3 black balls in an urn. Two balls are randomly extracted from the urn. What is the probability that both balls are white? 2/7
20. There are 4 white and 6 black balls in an urn. Two balls are randomly taken from the urn.
21. There are 5 children in a family. Assuming that probabilities of birth of boy and girl are equal, find the probability that the family has three boys: 5/16
22. There are 5 children in a family. Assuming that probabilities of birth of boy and girl are equal, find the probability that there are 3 girls and 2 boys in the family.
23. There are 9 white and 1 black balls in an urn. Three balls are randomly extracted from the urn. What is the probability that all balls are white? 7/10
24. Three dice are tossed. Find the probability that the sum of aces will be 5. 1/36
25. Three shooters shoot in a target. Probability of hit in the target by the 1st shooter is 0,75; by the 2^nd - 0,8 and by the 3^rd - 0,9. Find the probability of hit by all the shooters. 0,54
26. Two dice are tossed. Find the probability that the product of aces does not exceed 2. 1/12
27. Two dice are tossed. Find the probability that the sum of aces does not exceed 3. 1/12
28. Two dice are tossed. Find the probability that the sum of aces doesn't exceed 4. 1/6
29. Two dice are tossed. What is the probability that the sum of aces will be more than 10? 1/12
30. Two shooters shoot in a target. The probability of hit by the 1st shooter is 0,6, and by the 2nd - 0,7. Find the probability that at least one of shooters will hit in the target. 0,88
31. Two shooters shoot in a target. The probability of hit by the first shooter is 0,6, and by the second - 0,7. Find the probability that only one of shooters will hit in the target. 0,46
32. Two shots are made in a target by two shooters. The probability of hit by the first shooter is equal to 0,8, by the second - 0,9. Find the probability of at least one hit in the target. 0,98
33. We say that a discrete random variable X is distributed under a binomial law if P(X=K) =
34. We say that a discrete random variable X is distributed under a geometrical law if pq^n-1
35. We say that a discrete random variable X is distributed under Poisson law with parameter l if
36. What does the formula determine? Initial moment of the k order
37. What does the formula determine? Central moment of the k order of a variation series
38. What does the formula determine? Excess
39. What does the formula determine? Asymmetry

What is the probability that both balls are black? 1/3

1. Which estimator of mathematical expectation (parent mean) is sample mean ? Unbiased and consistent
2. Which estimator of parent dispersion is sample dispersion D_s? biased

Индекс цен переменного состава: Индекс цен постоянного состава:

Индекс cтруктуры:

Взаимосвязь между индексами:

б) индивидуальные и общие индексы

Индивидуальный индекс цен

рассчитывается по формуле:

Индивидуальный индекс физического объема рассчитаем по формуле:

 

Общий индекс цен рассчитывается по формуле:

Общий индекс физического объема рассчитывается по формуле:

 

Общий индекс товарооборота

рассчитывается по формулам:

Разложение абсолютного прироста по факторам можно записать в виде:

 

 

Индекс физического объема товарооборота:

 

Общий индекс товарооборота: , где