Final exam trial tests (for self-checking)
1. Let A and B be events connected with the same trial. Show the event that means occurrence of at least one of the events A and B.
A) ; B) АВ; C) A + B; D) ;
E) There is no right answer.
2. Let А1, А2, А3 be events connected with the same trial. Let A be the event that means occurrence of exactly two of the events А1, А2 and А3. Express the event A by the events А1, А2 and А3.
A) A = А1 А2 А3; B) A = А1А2 + А2А3 + А1 А3; C) A = А2А3; D) A = А1А3;
E) .
3. A coin is tossed twice. Find the probability that the coin lands on «tails» in both times.
A) 1/2; B) 3/4; C) 1; D) 1/4; E) There is no right answer.
4. Two dice are tossed. Find the probability that the product of aces does not exceed 5.
A) 0,5; B) 5/6; C) 1/6; D) 5/36; E) 5/18.
5. There are 10 white, 15 black, 20 blue and 25 red balls in an urn. One ball is randomly extracted. Find the probability that the extracted ball is blue or red.
A) 5/14; B) 1/70; C) 1/7; D) 9/14; E) 3/98.
6. There are 10 white, 15 black, 20 blue and 25 red balls in an urn. One ball is randomly extracted. Find the probability that the extracted ball is either white, either black or blue.
A) 5/14; B) 1/70; C) 1/7; D) 9/14; E) 3/98.
7. One letter is randomly chosen from the word "ECONOMIST". What is the probability that this letter is "A"?
A) 0,6; B) 1/6; C) 1/2; D) 0; E) There is no right answer.
8. There are 5 white and 5 black balls in an urn. What is the probability of extracting a red ball from the urn?
A) 1; B) 0; C) 1/2; D) 1/5; E) 1/3.
9. There are 10 white, 4 black, 12 yellow and 15 red balls in a box. Find the probability that a randomly taken ball will be black.
A) 1/13; B) 4/41; C) 5/14; D) 3/41; E) 1/14.
10. There are 10 white, 15 black, 10 yellow and 25 red balls in a box. Find the probability that a randomly taken ball will be white.
A) 2/3; B) 1/7; C) 3/14; D) 1/6; E) 1/14.
11. Three coins are tossed. What is the probability that at least one of the coins lands on «heads»?
A) 1/8; B) 7/8; C) 1/4; D) 3/4; E) 1/3.
12. Three shooters shoot in a target. The probability of hit in the target at one shot for the 1st shooter is 0,75; for the 2nd – 0,8 and for the 3rd – 0,2. Find the probability that at least one of the shooters will hit in the target.
A) 0,995; B) 0,96; C) 0,95; D) 0,65; E) 0,504.
13. Two dice are tossed. Find the probability that the sum of aces is divided on 5.
A) 1/7; B) 1/4; C) 7/36; D) 1/6; E) There is no right answer.
14. The probability of delay for the train № 1 is equal to 0,2, and for the train № 2 – 0,64. Find the probability that only one train will be late.
A) 0,615; B) 0,584; C) 0,715; D) 0,9; E) There is no right answer.
15. The events A and B are independent, Р(А) = 0,3; Р(В) = 0,1. Find .
A) ; B) ; C) ;
D) ; E) There is no right answer.
16. The probability of distorting a sign at transmitting a message is equal to 0,01. Find the probability that a transmitted message consisting of 10 signs contains exactly 3 distortions (distortion – искажение).
A) ; B) ; C) 0,013; D) ;
E) There is no right answer.
17. Two shots are made in a target by two guns. The probability of hit from the first gun is 0,8, from the second gun – 0,9. Find the probability of only one hit in the target.
A) 0,2; B) 0,17; C) 0,26; D) 0,72; E) 0,5.
18. The law of distribution of a discrete random variable X is given by the following table:
| X | –1 | Y |
| P | 0.4 | 0.6 |
It is known that M (X) = 2,6. Find Y.
A) 1; B) 2; C) 3; D) 4; E) 5.
19. A discrete random variable X is given by the following law of distribution:
| X | -5 | ||
| P | 0,1 | 0,6 | 0,2 |
Find the mathematical expectation M(X).
A) 0,5; B) 2,5; C) 1; D) 10; E) 3.
20. A discrete random variable X is given by the following law of distribution:
| X | x1 | x2 | x3 | x4 |
| P | 0,2 | 0,1 | p3 | 0.3 |
Find p3.
A) 0,5; B) 0,2; C) 0; D) 1; E) 0,4.
21. The dispersion of the number of occurrences of an event A for n independent trials in each of which the probability of occurrence of the event A is p, is equal to:
A) ; B) ; C) ; D) ;
E) .
22. The random variables X and Y are independent. Find the dispersion of the random variable if it is known that .
A) 30; B) 12; C) 78; D) 114; E) 110.
23. If F(x) is an integral function of distribution of a random variable , then for x > b F(x) is equal to:
A) 1; B) 0; C) 0,5; D) – 1; E) 2.
24. If all possible values of a random variable X belong to an interval (a, b) then the integral function of distribution F(x) for x £ a is equal to:
A) 1; B) 0; C) 0,5; D) – 1; E) 2.
25. The graph of the density of distribution f(x) of a continuous random variable X cannot be:
A) below the axis Ox; B) upper the axis Ox; C) to the left of the axis Оу;
D) to the right of the axis Оу; E) There is no right answer.
26. If a continuous random variable X is uniformly distributed over (a, b), then the density has the following form:
A) B)
C) D)
Е)
27. A random variable X has the following density of distribution:
.
Find the mathematical expectation M(X) and the dispersion D(X).
A) M(X) = – 3, D(X) = 3; B) M(X) = 3, D(X) = 3; C) M(X) = – 3, D(X) = 9;
D) M(X) = 0, D(X) = 1; E) There is no right answer.
28. Write the density of probability of a normally distributed random variable X if M(X) = 5, D(X) = 16.
A) ; B) ; C) ; D) ;
E) .
29. An urn contains 5 red, 3 white, and 4 blue balls. What is the probability of extracting a black ball from the urn?
A) 1/3; B) 0; C) 0,25; D) 0,5; E) 5/12.
30. An urn contains 3 white and 7 black balls. Two balls are randomly extracted from the urn. What is the probability that both balls are white?
A) 6/15; B) 7/15; C) 7/10; D) 8/15; E) 1/15.
31. In a batch of 5 products 2 of them have the first grade, and 3 – the second grade. Two products are randomly taken. Find the probability that both products have the second grade.
A) 0,6; B) 0,3; C) 0,1; D) 0,5; E) 0,4.
32. In a batch of 5 products 2 of them have the first grade, and 3 – the second grade. Two products are randomly taken. Find the probability that they have different grades.
A) 0,6; B) 0,3; C) 0,1; D) 0,5; E) 0,4.
33. Two shooters simultaneously shoot in a target. The probability of hit in the target by the first shooter is 0,7, and by the second – 0,8. Find the probability that none of shooters will hit in the target.
A) 0,38; B) 0,94; C) 0,06; D) 0,56; E) 0,96.
34. A coin is tossed 6 times. Find the probability that the coin lands on heads only once.
A) 7/64; B) 15/64; C) 57/64; D) 21/32; E) 3/32.
35. A coin is tossed 6 times. Find the probability that the coin lands on heads 6 times.
A) 7/64; B) 15/64; C) 1/64; D) 21/32; E) 3/32.
36. A coming up seeds of wheat makes 80 %. Find the probability that two of five sown seeds will come up.
A) 0,496; B) 0,0064; C) 0,0512; D) 0,00032; E) 0,2048.
37. A coming up seeds of wheat makes 80 %. Find the probability that none of five sown seeds will come up.
A) 0,00032; B) 0,0064; C) 0,0512; D) 0,496; E) 0,2048.
38. During the year two firms have an opportunity independently from each other to go bankrupt with probabilities 0,05 and 0,1. Find the probability that at the end of the year both firms will function.
A) 0,64; B) 0,855; C) 0,45; D) 0,38; E) 0,256.
39. A shooter makes 3 shots in a target. The probability of hit in the target at one shot is equal to 0,7. Find the probability that at least one shot will hit in the target.
A) 0,973; B) 0,027; C) 0,316; D) 0,36; E) 0,3.
40. Find the mathematical expectation M(X) of a random variable X, knowing its law of distribution:
| xi | |||
| pi | 0,2 | 0,3 | 0,5 |
A) 3,4; B) 2,8; С) 2,6; D) 2,4; E) 0,76.
41. Find the dispersion D(X) of a random variable X, knowing its law of distribution:
| x i | |||
| pi | 0,3 | 0,2 | 0,5 |
А) 0,76; B) 2,6; С) 2,8; D) 2,4; E) 3,4.
42. An integral function of distribution F(х) is:
A) decreasing; B) periodic; C) even; D) non-decreasing; E) odd.
43. A random variable X is given by the integral function of distribution:
Find the mathematical expectation M(X).
A) 1/3; B) 1; C) 2/3; D) 0,5; E) 0.
44. A random variable X is given by the integral function of distribution:
Find the dispersion D(X).
A) 1/3; B) 1; C) 2/3; D) 0,5; E) 1/12.
45. A random variable X is given by the integral function of distribution:
Calculate the probability of hit of the random variable X in the interval (0; 2).
A) 0,5; B) 0; C) 1/4; D) 1; E) 2/3.
46. A random variable X has the following law of distribution:
| xi | ||||
| pi | 1/30 | 3/10 | ½ | 1/6 |
Find the mathematical expectation of X.
A) 1; B) 1,5; C) 2; D) 1,8; E) 2,3.
47. Let a random variable X be given by the following law of distribution:
| xi | – 3 | – 2 | |||
| pi | 0,1 | 0,2 | 0,4 | 0,2 | 0,1 |
Find the mathematical expectation M(X).
A) 2; B) 0,1; C) 0,2; D) 0; E) 1
48. Let a random variable X be given by the following law of distribution:
| xi | -3 | -2 | |||
| pi | 0,1 | 0,2 | 0,4 | 0,2 | 0,1 |
Find the dispersion D(X).
A) 0; B) 3,4; C) 1; D) 0,34; E) 0,1.
49. A random variable X is given by the integral function of distribution:
Find the probability of hit of the random variable X into the interval (2; 3).
A) 0,25; B) 0,5; C) 1/3; D) 2/3; E) 1.
50. Show one of the properties of mathematical expectation (C is a constant):
A) ; B) ; C) ;
D) ; E)
51. The sample is given by the following table:
| xi | ||||
| ni |
Find the sample mean .
A) 5; B) 6; C) 5,76; D) 4,65; E) There is no right answer.
52. The sample has been taken from a parent population:
| xi | ||||
| ni |
Find the unbiased estimation of the parent mean (the sample mean ).
A) 18; B) 4; C) 3: D) 6; E) 30.
53. The sample is given by the following table:
| xi | –20 | |||
| ni |
Find the sample dispersion s2.
A)167,29; B) 152,25; C)145,5; D)182,35; E)175,43.
54. The sample is given by the following table:
| xi | 0,5 | –2 | |
| ni |
Find the unbiased estimation of the dispersion s2 (the corrected dispersion).
A) 5; B) 6; C) 10,95; D) 4,65; E) 1.
54. The unbiased estimation of the dispersion found on a sample of volume n = 41 is = 3. Find the displaced estimation of the dispersion s2.
A) 4,05; B) 7,8; C) 5,3; D) 2,93; E) 20.
55. Find the mode of the following variation series:
| xi | ||||
| ni |
A) 10; B) 11,5; C) 7; D) 3; E) 1.
56. The distribution of an attribute X – the number of transactions at stock exchange for a quarter received on n = 400 observations is given:
| xi | |||||||||||
| ni |
Find the arithmetic mean .
A) ; B) ; C) ; D) ; E) .
57. Find the mode of the following sample: {10, 10, 15, 16, 16, 16, 16, 16}.
A) 16; B) 10; C) 20; D) 5; E) 125.
58. Find the median of the following sample: {10, 10, 15, 16, 16, 16, 16}.
A) 16; B) 10; C) 25; D) 15; E) 148.
59. Results of sampling inspections of one-day profits of city shops are given: {5, 5, 10, 14, 18, 20, 25}. Find the median.
A) 30; B) 10; C) 25; D) 14; E) 148.
60. How will the average value change if all variants are increased in 5 times?
A) will increase in 5 times; B) will decrease in 5 times;
C) will increase in 25 times; D) will decrease in 25 times; E) will not change.
61. How will the sample dispersion if all variants are increased on 6?
A) will decrease on 36; B) will increase on 6; C) will increase in 6 times;
D) will decrease in 6 times; E) will not change.
62. How will the length of confidence interval if the volume of a sample will increase in 9 times?
A) will decrease in 3 times; B) will increase in 3 times;
C) will decrease in 9 times; D) will increase in 9 times; E) will not change.
63. A discrete variation series of a sample is given:
| x i | –2 | ||
| ni |
Find the volume of the variation series:
A) 3; B) 25; C) 16; D) 4; E) 1.
64. Find the confidence interval for an estimation with reliability = 0,95 of the parent mean of a normally distributed attribute X if the parent average quadratic deviation s = 9, the sample mean = 18,31, the volume of a sample n = 49. It has been foundon the table: t = 1,96.
A) (15,79; 20,83); B) (17,138; 19,882); C) (16,23; 18,65); D) (14,83; 17,12);
E) (15,8; 21,02).
65. What is the mode of a normal distribution with mathematical expectation 0 and dispersion 1?
A) е; B) 0; C) 1/2; D) –1/2; E) 1.
66. A discrete variation series is given:
| xi | –5 | ||
| ni |
Find the arithmetic mean.
A) 3; B) 4; C) 0,4; D) 3,5; E) 2.
67. The sample is given as the following distribution of frequencies:
| xi | |||
| ni |
Find the sample mean.
A) 5,9; B) 4; C) 3; D) 4,5; E) 3,5.
68. The variation series of a sample is given:
| xi | ||||
| ni |
Find the sample dispersion s2.
A) 2; B) 1,8; C) 4; D) 5; E) 120.
69. A random variable X is distributed under a Poisson law. Find by the method of moments a point estimation of unknown parameter of the Poisson distribution on the following sample:
| xi | |||||||
| ni |
A) 2,5; B) 4; C) 3,5; D) 0,59; E) 0,9.
70. A random variable X is distributed under a normal law. Find by the method of maximal plausibility a point estimation of unknown parameter a of the normal distribution on the following sample:
| xi | |||
| ni |
A) 3,3; B) 4; C) 0,5; D) 0,59; E) 0,9.
APPENDIX
Values the functions and
| x | x | x | ||||||||||||||
| 0,00 | 0,3989 | 0,0000 | 0,36 | 0,3739 | 0,1406 | 0,72 | 0,3079 | 0,2642 | ||||||||
| 0,01 | 0,3989 | 0,0040 | 0,37 | 0,3726 | 0,1443 | 0,73 | 0,3056 | 0,2673 | ||||||||
| 0,02 | 0,3989 | 0,0080 | 0,38 | 0,3712 | 0,1480 | 0,74 | 0,3034 | 0,2703 | ||||||||
| 0,03 | 0,3988 | 0,0120 | 0,39 | 0,3697 | 0,1516 | 0,75 | 0,3011 | 0,2734 | ||||||||
| 0,04 | 0,3986 | 0,0160 | 0,40 | 0,3683 | 0,1554 | 0,76 | 0,2989 | 0,2764 | ||||||||
| 0,05 | 0,3984 | 0,0199 | 0,41 | 0,3668 | 0,1591 | 0,77 | 0,2966 | 0,2794 | ||||||||
| 0,06 | 0,3982 | 0,0239 | 0,42 | 0,3652 | 0,1628 | 0,78 | 0,2943 | 0,2823 | ||||||||
| 0,07 | 0,3980 | 0,0279 | 0,43 | 0,3637 | 0,1664 | 0,79 | 0,2920 | 0,2852 | ||||||||
| 0,08 | 0,3977 | 0,0319 | 0,44 | 0,3621 | 0,1700 | 0,80 | 0,2897 | 0,2881 | ||||||||
| 0,09 | 0,3973 | 0,0359 | 0,45 | 0,3605 | 0,1736 | 0,81 | 0,2874 | 0,2910 | ||||||||
| 0,10 | 0,3970 | 0,0398 | 0,46 | 0,3589 | 0,1772 | 0,82 | 0,2850 | 0,2939 | ||||||||
| 0,11 | 0,3965 | 0,0438 | 0,47 | 0,3572 | 0,1808 | 0,83 | 0,2827 | 0,2967 | ||||||||
| 0,12 | 0,3961 | 0,0478 | 0,48 | 0,3555 | 0,1844 | 0,84 | 0,2803 | 0,2995 | ||||||||
| 0,13 | 0,3956 | 0,0517 | 0,49 | 0,3538 | 0,1879 | 0,85 | 0,2780 | 0,3023 | ||||||||
| 0,14 | 0,3951 | 0,0557 | 0,50 | 0,3521 | 0,1915 | 0,86 | 0,2756 | 0,3051 | ||||||||
| 0,15 | 0,3945 | 0,0596 | 0,51 | 0,3503 | 0,1950 | 0,87 | 0,2732 | 0,3078 | ||||||||
| 0,16 | 0,3939 | 0,0636 | 0,52 | 0,3485 | 0,1985 | 0,88 | 0,2709 | 0,3106 | ||||||||
| 0,17 | 0,3932 | 0,0675 | 0,53 | 0,3467 | 0,2019 | 0,89 | 0,2685 | 0,3133 | ||||||||
| 0,18 | 0,3925 | 0,0714 | 0,54 | 0,3448 | 0,2054 | 0,90 | 0,2661 | 0,3159 | ||||||||
| 0,19 | 0,3918 | 0,0753 | 0,55 | 0,3429 | 0,2088 | 0,91 | 0,2637 | 0,3186 | ||||||||
| 0,20 | 0,3910 | 0,0793 | 0,56 | 0,3410 | 0,2123 | 0,92 | 0,2613 | 0,3212 | ||||||||
| 0,21 | 0,3902 | 0,0832 | 0,57 | 0,3391 | 0,2157 | 0,93 | 0,2589 | 0,3238 | ||||||||
| 0,22 | 0,3894 | 0,0871 | 0,58 | 0,3372 | 0,2190 | 0,94 | 0,2565 | 0,3264 | ||||||||
| 0,23 | 0,3885 | 0,0910 | 0,59 | 0,3352 | 0,2224 | 0,95 | 0,2541 | 0,3289 | ||||||||
| 0,24 | 0,3876 | 0,0948 | 0,60 | 0,3332 | 0,2257 | 0,96 | 0,2516 | 0,3315 | ||||||||
| 0,25 | 0,3867 | 0,0987 | 0,61 | 0,3312 | 0,2291 | 0,97 | 0,2492 | 0,3340 | ||||||||
| 0,26 | 0,3857 | 0,1026 | 0,62 | 0,3292 | 0,2324 | 0,98 | 0,2468 | 0,3365 | ||||||||
| 0,27 | 0,3847 | 0,1064 | 0,63 | 0,3271 | 0,2357 | 0,99 | 0,2444 | 0,3389 | ||||||||
| 0,28 | 0,3836 | 0,1103 | 0,64 | 0,3251 | 0,2389 | 1,00 | 0,2420 | 0,3413 | ||||||||
| 0,29 | 0,3825 | 0,1141 | 0,65 | 0,3230 | 0,2422 | 1,01 | 0,2396 | 0,3438 | ||||||||
| 0,30 | 0,3814 | 0,1179 | 0,66 | 0,3209 | 0,2454 | 1,02 | 0,2371 | 0,3461 | ||||||||
| 0,31 | 0,3802 | 0,1217 | 0,67 | 0,3187 | 0,2486 | 1,03 | 0,2347 | 0,3485 | ||||||||
| 0,32 | 0,3790 | 0,1255 | 0,68 | 0,3166 | 0,2517 | 1,04 | 0,2323 | 0,3508 | ||||||||
| 0,33 | 0,3778 | 0,1293 | 0,69 | 0,3144 | 0,2549 | 1,05 | 0,2299 | 0,3531 | ||||||||
| 0,34 | 0,3765 | 0,1331 | 0,70 | 0,3123 | 0,2580 | 1,06 | 0,2275 | 0,3554 | ||||||||
| 0,35 | 0,3752 | 0,1368 | 0,71 | 0,3101 | 0,2611 | 1,07 | 0,2251 | 0,3577 | ||||||||
| x | x | x | ||||||||||||||
| 1,08 | 0,2227 | 0,3599 | 1,49 | 0,1315 | 0,4319 | 1,90 | 0,0656 | 0,4713 | ||||||||
| 1,09 | 0,2203 | 0,3621 | 1,50 | 0,1295 | 0,4332 | 1,91 | 0,0644 | 0,4719 | ||||||||
| 1,10 | 0,2179 | 0,3643 | 1,51 | 0,1276 | 0,4345 | 1,92 | 0,0632 | 0,4726 | ||||||||
| 1,11 | 0,2155 | 0,3665 | 1,52 | 0,1257 | 0,4357 | 1,93 | 0,0620 | 0,4732 | ||||||||
| 1,12 | 0,2131 | 0,3686 | 1,53 | 0,1238 | 0,4370 | 1,94 | 0,0608 | 0,4738 | ||||||||
| 1,13 | 0,2107 | 0,3708 | 1,54 | 0,1219 | 0,4382 | 1,95 | 0,0596 | 0,4744 | ||||||||
| 1,14 | 0,2083 | 0,3729 | 1,55 | 0,1200 | 0,4394 | 1,96 | 0,0584 | 0,4750 | ||||||||
| 1,15 | 0,2059 | 0,3749 | 1,56 | 0,1182 | 0,4406 | 1,97 | 0,0573 | 0,4756 | ||||||||
| 1,16 | 0,2036 | 0,3770 | 1,57 | 0,1163 | 0,4418 | 1,98 | 0,0562 | 0,4761 | ||||||||
| 1,17 | ,2012 | 0,3790 | 1,58 | 0,1145 | 0,4429 | 1,99 | 0,0551 | 0,4767 | ||||||||
| 1,18 | 0,1989 | 0,3810 | 1,59 | 0,1127 | 0,4441 | 2,00 | 0,0540 | 0,4772 | ||||||||
| 1,19 | 0,1965 | 0,3830 | 1,60 | 0,1109 | 0,4452 | 2,02 | 0,0519 | 0,4783 | ||||||||
| 1,20 | 0,1942 | 0,3849 | 1,61 | 0,1092 | 0,4463 | 2,04 | 0,0498 | 0,4793 | ||||||||
| 1,21 | 0,1919 | 0,3869 | 1,62 | 0,1074 | 0,4474 | 2,06 | 0,0478 | 0,4803 | ||||||||
| 1,22 | 0,1895 | 0,3883 | 1,63 | 0,1057 | 0,4484 | 2,08 | 0,0459 | 0,4812 | ||||||||
| 1,23 | 0,1872 | 0,3907 | 1,64 | 0,1040 | 0,4495 | 2,10 | 0,0440 | 0,4821 | ||||||||
| 1,24 | 0,1849 | 0,3925 | 1,65 | 0,1023 | 0,4505 | 2,12 | 0,0422 | 0,4830 | ||||||||
| 1,25 | 0,1826 | 0,3944 | 1,66 | 0,1006 | 0,4515 | 2,14 | 0,0404 | 0,4838 | ||||||||
| 1,26 | 0,1804 | 0,3962 | 1,67 | 0,0989 | 0,4525 | 2,16 | 0,0387 | 0,4846 | ||||||||
| 1,27 | 0,1781 | 0,3980 | 1,68 | 0,0973 | 0,4535 | 2,18 | 0,0371 | 0,4854 | ||||||||
| 1,28 | 0,1758 | 0,3997 | 1,69 | 0,0957 | 0,4545 | 2,20 | 0,0355 | 0,4861 | ||||||||
| 1,29 | 0,1736 | 0,4015 | 1,70 | 0,0940 | 0,4554 | 2,22 | 0,0339 | 0,4868 | ||||||||
| 1,30 | 0,1714 | 0,4032 | 1,71 | 0,0925 | 0,4564 | 2,24 | 0,0325 | 0,4875 | ||||||||
| 1,31 | 0,1691 | 0,4049 | 1,72 | 0,0909 | 0,4573 | 2,26 | 0,0310 | 0,4881 | ||||||||
| 1,32 | 0,1669 | 0,4066 | 1,73 | 0,0893 | 0,4582 | 2,28 | 0,0297 | 0,4887 | ||||||||
| 1,33 | 0,1647 | 0,4082 | 1,74 | 0,0878 | 0,4591 | 2,30 | 0,0283 | 0,4893 | ||||||||
| 1,34 | 0,1626 | 0,4099 | 1,75 | 0,0863 | 0,4599 | 2,32 | 0,0270 | 0,4898 | ||||||||
| 1,35 | 0,1604 | 0,4115 | 1,76 | 0,0848 | 0,4608 | 2,34 | 0,0258 | 0,4904 | ||||||||
| 1,36 | 0,1582 | 0,4131 | 1,77 | 0,0833 | 0,4616 | 2,36 | 0,0246 | 0,4909 | ||||||||
| 1,37 | 0,1561 | 0,4147 | 1,78 | 0,0818 | 0,4625 | 2,38 | 0,0235 | 0,4913 | ||||||||
| 1,38 | 0,1539 | 0,4162 | 1,79 | 0,0804 | 0,4633 | 2,40 | 0,0224 | 0,4918 | ||||||||
| 1,39 | 0,1518 | 0,4177 | 1,80 | 0,0790 | 0,4641 | 2,42 | 0,0213 | 0,4922 | ||||||||
| 1,40 | 0,1497 | 0,4192 | 1,81 | 0,0775 | 0,4649 | 2,44 | 0,0203 | 0,4927 | ||||||||
| 1,41 | 0,1476 | 0,4207 | 1,82 | 0,0761 | 0,4656 | 2,46 | 0,0194 | 0,4931 | ||||||||
| 1,42 | 0,1456 | 0,4222 | 1,83 | 0,0748 | 0,4664 | 2,48 | 0,0184 | 0,4934 | ||||||||
| 1,43 | 0,1435 | 0,4236 | 1,84 | 0,0734 | 0,4671 | 2,50 | 0,0175 | 0,4938 | ||||||||
| 1,44 | 0,1415 | 0,4251 | 1,85 | 0,0721 | 0,4678 | 2,52 | 0,0167 | 0,4941 | ||||||||
| 1,45 | 0,1394 | 0,4265 | 1,86 | 0,0707 | 0,4686 | 2,54 | 0,0158 | 0,4945 | ||||||||
| 1,46 | 0,1374 | 0,4279 | 1,87 | 0,0694 | 0,4693 | 2,56 | 0,0151 | 0,4948 | ||||||||
| 1,47 | 0,1354 | 0,4292 | 1,88 | 0,0681 | 0,4699 | 2,58 | 0,0143 | 0,4951 | ||||||||
| 1,48 | 0,1334 | 0,4306 | 1,89 | 0,0669 | 0,4706 | 2,60 | 0,0136 | 0,4953 | ||||||||
| x | x | x | ||||||||||||||
| 2,62 | 0,0129 | 0,4956 | 2,88 | 0,0063 | 0,4980 | |||||||||||
| 2,64 | 0,0122 | 0,4959 | 2,90 | 0,0060 | 0,4981 | 3,50 | 0,00087 | 0,49977 | ||||||||
| 2,66 | 0,0116 | 0,4961 | 2,92 | 0,0056 | 0,4982 | |||||||||||
| 2,68 | 0,0110 | 0,4963 | 2,94 | 0,0053 | 0,4984 | 3,60 | 0,00061 | 0,499841 | ||||||||
| 2,70 | 0,0104 | 0,4965 | 2,96 | 0,0050 | 0,4985 | 3,70 | 0,00042 | 0,49989 | ||||||||
| 2,72 | 0,0099 | 0,4967 | 2,98 | 0,0047 | 0,4986 | 3,80 | 0,00029 | 0,499928 | ||||||||
| 2,74 | 0,0093 | 0,4969 | 3,00 | 0,00443 | 0,49865 | |||||||||||
| 2,76 | 0,0088 | 0,4971 | 3,90 | 0,00020 | 0,49995 | |||||||||||
| 2,78 | 0,0084 | 0,4973 | 3,10 | 0,00327 | 0,49903 | 4,00 | 0,0001338 | 0,499968 | ||||||||
| 2,80 | 0,0079 | 0,4974 | 3,20 | 0,00238 | 0,49931 | |||||||||||
| 2,82 | 0,0075 | 0,4976 | 4,50 | 0,0000160 | 0,499997 | |||||||||||
| 2,84 | 0,0071 | 0,4977 | 3,30 | 0,00172 | 0,49952 | 5,00 | 0,0000015 | 0,49999997 | ||||||||
| 2,86 | 0,0067 | 0,4979 | 3,40 | 0,00123 | 0,49966 | |||||||||||
LIST OF THE USED BOOKS
1. Гмурман В.Е., Теория вероятностей и математическая статистика. Издание седьмое, стереотипное, Москва: «Высшая школа», 2000.
2. Сборник задач. Математика для экономистов. Под редакцией профессора Кельтеновой Р.Т., Алматы: «Экономика», 2000.
3. Горелова Г.В., Кацко И.А., Теория вероятностей и математическая статистика в примерах и задачах с применением Excel. – Издание четвертое, Ростов на Дону: «Феникс», 2006.
4. Коршунов Д.А., Фосс С.Г., Сборник задач и упражнений по теории вероятностей, Новосибирск, 1997.
5. Коршунов Д.А., Чернова Н.И., Сборник задач и упражнений по математической статистике, Новосибирск. 2001.
6. Кремер Н.Ш., Теория вероятностей и математическая статистика: Учебник для ВУЗов. — 2-е изд., перераб. и доп. М: ЮНИТИ-ДАНА, 2004. — 573 с.
7. Данко П.Е., Попов А.Г., Кожевникова Т.Я., Высшая математика в упражнениях и задачах. Издание пятое, исправленное. – Москва: «Высшая школа», 1999, часть 2.
8. Манита А.Д., Теория вероятностей и математическая статистика: Учебное пособие. – М.: Издательский отдел УНЦ ДО, 2001, 120 с.
9. Гмурман В.Е., Руководство к решению задач по теории вероятностей и математической статистике: Учебное пособие — 11-е изд., перераб. — М.: Высшее образование, 2006, 404 с.
10. Olav Kallenberg, Foundations of Modern Probability, second edition, Springer Series in Statistics, 2002, 650 pages.
11. Olav Kallenberg, Probabilistic Symmetries and Invariance Principles, Springer-Verlag, New York, 2005, 510 pages.
12. Hopkins W.G., A New View of Statistics, 2000, Internet Society for Sport Science: http://www.sportsci.org/resource/stats/.
13. Sheldon Ross, A First Course in Probability, seventh edition, University of Southern California, Pearson Education International, 2006, 561 pages.
CONTENTS
| INTRODUCTION……………………………………………………….. | |
| LECTURE 1. Basic concepts of probability theory……………………….. | |
| Classical definition of probability ………………………………………….. | |
| Relative frequency …………………………………………………………. | |
| Geometric probabilities …………………………………………………….. | |
| Exercises ……………………………………………………………........... | |
| LECTURE 2. Basic formulas of combinatorial analysis ……………..…… | |
| Operations over events …………………………………..………………..… | |
| Exercises …………………………………………………………………… | |
| LECTURE 3.Theorem of addition of probabilities of incompatible events | |
| Complete group of events …………………………………………..……….. | |
| Opposite events………………………………………………………………. | |
| Conditional probability ……………………………………………………… | |
| Theorem of multiplication of probabilities…………………………………… | |
| Exercises …………………………………………………………………… | |
| LECTURE 4.Independent events …………………………………………. | |
| Probability of appearance of at least one event …………………..………… | |
| Exercises ………………………………………………………..…………… | |
| LECTURE 5. Theorem of addition of probabilities of compatible events … | |
| Formula of total probability………………………………………………… | |
| Probability of hypotheses. Bayes’s formulas……………………………….. | |
| Exercises …………………………………………………………………… | |
| LECTURE 6. Repetition (recurrence) of trials. The Bernoulli formula…… | |
| Local theorem of Laplace…………………………………..………… | |
| Integral theorem of Laplace ………………………………………………… | |
| Exercises ……………………………………………………………..……… | |
| LECTURE 7. Random variables. The law of distribution of a discrete random variable …………………………………………………………….. | |
| Mathematical operations over random variables……………………………. | |
| (Mathematical) expectation of a discrete random variable………………….. | |
| Dispersion of a discrete random variable…………………………………… | |
| Exercises ………………………………………………………………..…… | |
| LECTURE 8. Distribution function of a random variable………………….. | |
| Properties of a distribution function…………………..………………..…… | |
| Continuous random variables. Probability density………………………….. | |
| Properties of probability density…………………………………………….. | |
| Exercises …………………………………………………………………... | |
| LECTURE 9. Basic laws of distribution of discrete random variables…….. | |
| Binomial law of distribution…………………………………………………. | |
| The law of distribution of Poisson……………………………………………. | |
| Geometric distribution ………………………………………..……………… | |
| Hypergeometric distribution………………………………………………….. | |
| Exercises …………………………………………………………..…….…... | |
| LECTURE 10. Basic laws of distribution of continuous random variables…. | |
| Uniform law of distribution ……………………………………………......... | |
| Exponential law of distribution………………………………………………. | |
| Normal law of distribution……………………………………………………. | |
| Exercises ……………………………………………………………………. | |
| LECTURE 11.The law of large numbers and limit theorems………………. | |
| The central limit theorem…………………………………………………….. | |
| Exercises ……………………………………………………………............ | |
| LECTURE 12.Mathematical statistics. Variation series and their characteristics…………………………………………………………………. | |
| Numerical characteristics of variation series…………………………………. | |
| Exercises …………………………………………………………………..... | |
| LECTURE 13.Bases of the mathematical theory of sampling…………….. | |
| Exercises …………………………………………………………………….. | |
| LECTURE 14. Methods of finding of estimations…………………………. | |
| Method of moments …………………………………………………………. | |
| Method of maximal (the greatest) plausibility………………………………. | |
| Method of the least squares…………………………………………………. | |
| Notion of interval estimation……………………………………………….. | |
| Exercises ……………………………………………………………………. | |
| LECTURE 15. Testing of statistical hypotheses…………………………… | |
| Exercises ………………………………………………………………......... INDIVIDUAL HOME WORKS………………………………………… FINAL EXAM TRIAL TESTS (FOR SELF-CHECKING)……………. | |
| APPENDIX…………………………………………………………………. | |
| LIST OF THE USED BOOKS…………………………………………… |