Davydov

Ɇɨɫɤɨɜɫɤɢɣ ɚɜɬɨɦɨɛɢɥɶɧɨ-ɞɨɪɨɠɧɵɣ ɢɧɫɬɢɬɭɬ (Ƚɨɫɭɞɚɪɫɬɜɟɧɧɵɣ ɬɟɯɧɢɱɟɫɤɢɣ ɭɧɢɜɟɪɫɢɬɟɬ)

Ʉɚɮɟɞɪɚ ɜɵɫɲɟɣ ɦɚɬɟɦɚɬɢɤɢ

ȿ.Ƚ.Ⱦɚɜɵɞɨɜ

Ʌɚɛɨɪɚɬɨɪɧɵɣ ɩɪɚɤɬɢɤɭɦ

ɩɨ ɫɩɟɰɢɚɥɶɧɵɦ ɝɥɚɜɚɦ ɜɵɫɲɟɣ ɦɚɬɟɦɚɬɢɤɢ

Ⱦɥɹ ɫɬɭɞɟɧɬɨɜ 2-3 ɤɭɪɫɨɜ ɢ ɚɫɩɢɪɚɧɬɨɜ

Ɇɟɬɨɞɢɱɟɫɤɢɟ ɭɤɚɡɚɧɢɹ

Ɇɨɫɤɜɚ

2004

Ʌɚɛɨɪɚɬɨɪɧɵɟ ɪɚɛɨɬɵ ɢ ɦɟɬɨɞɢɱɟɫɤɢɟ ɭɤɚɡɚɧɢɹ

ɩɨ ɫɩɟɰɢɚɥɶɧɵɦ ɝɥɚɜɚɦ ɜɵɫɲɟɣ ɦɚɬɟɦɚɬɢɤɢ

ɋ ɩɨɦɨɳɶɸ ɦɧɨɝɨɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɩɪɨɝɪɚɦɦɧɨɝɨ ɩɚɤɟɬɚ Scientific WorkPlace, ɬɟɤɫɬɨɜɨɣ ɪɟɞɚɤɬɨɪ ɤɨɬɨɪɨɝɨ ɩɨɡɜɨɥɹɟɬ ɫɨɡɞɚɜɚɬɶ ɫɥɨɠɧɵɟ ɧɚɭɱɧɨ-ɬɟɯɧɢɱɟɫɤɢɟ ɞɨɤɭɦɟɧɬɵ, ɚ ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɩɪɨ- ɰɟɫɫɨɪ ɩɨɦɨɝɚɟɬ ɜɵɩɨɥɧɹɬɶ ɧɟɨɛɯɨɞɢɦɵɟ ɜɵɱɢɫɥɟɧɢɹ ɢ ɩɨɫɬɪɨɟɧɢɟ ɝɪɚɮɢɤɨɜ ɢ ɞɢɚɝɪɚɦɦ, ɦɨɠɧɨ ɨɪɝɚɧɢɡɨɜɚɬɶ ɜɵɱɢɫɥɢɬɟɥɶɧɵɣ ɩɪɚɤ- ɬɢɤɭɦ ɫ ɫɨɫɬɚɜɥɟɧɢɟɦ ɢɧɞɢɜɢɞɭɚɥɶɧɵɯ ɡɚɞɚɧɢɣ ɞɥɹ ɥɚɛɨɪɚɬɨɪɧɵɯ ɪɚɛɨɬ, ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɛɵɬɶ ɪɚɫɩɟɱɚɬɚɧɵ ɫ ɯɨɪɨɲɢɦ ɤɚɱɟɫɬɜɨɦ ɩɟɱɚ- ɬɢ. ɉɪɢ ɠɟɥɚɧɢɢ ɦɨɠɧɨ ɩɪɨɜɨɞɢɬɶ ɩɪɨɜɟɪɤɭ ɪɟɡɭɥɶɬɚɬɨɜ ɪɚɛɨɬɵ ɢ ɨɰɟɧɢɜɚɬɶ ɟɟ ɫ ɩɨɦɨɳɶɸ Exam Builder ɱɚɫɬɶɸ ɩɚɤɟɬɚ.

Ɋɚɫɫɦɨɬɪɢɦ ɩɪɢɦɟɪɵ ɧɟɤɨɬɨɪɵɯ ɢɡ ɬɚɤɢɯ ɪɚɛɨɬ, ɫɨɩɪɨɜɨɠɞɚɹ ɤɚɠɞɭɸ ɪɚɛɨɬɭ ɩɪɢɦɟɪɨɦ ɟɟ ɜɵɩɨɥɧɟɧɢɹ.

1.Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ 1. ɉɟɪɜɢɱɧɚɹ ɨɛɪɚɛɨɬɤɚ ɪɟɡɭɥɶɬɚɬɨɜ ɧɚɛɥɸɞɟɧɢɣ

1)ɋɨɫɬɚɜɢɬɶ ɜɵɛɨɪɤɭ ɨɛɴɟɦɨɦ q = 81 + P (P = Q mod 9 + Q mod 7, Q ɧɨɦɟɪ ɫɬɭɞɟɧɬɚ ɩɨ ɫɩɢɫɤɭ ɝɪɭɩɩɵ) ɢɡ ɧɨɪɦɚɥɶ- ɧɨɣ ɝɟɧɟɪɚɥɶɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ [ ɫ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɨɠɢɞɚ- ɧɢɟɦ = P ɢ ɫɪɟɞɧɢɦ ɤɜɚɞɪɚɬɢɱɟɫɤɢɦ ɨɬɤɥɨɧɟɧɢɟɦ = 3.

2)ɇɚɣɬɢ ɨɰɟɧɤɭ {¯ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ ɢ ɢɫɩɪɚɜɥɟɧɧɵɟ ɨɰɟɧɤɢ v2 ɢ v ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɞɢɫɩɟɪɫɢɢ 2 ɢ ɫɪɟɞɧɟɝɨ ɤɜɚɞɪɚ- ɬɢɱɟɫɤɨɝɨ ɨɬɤɥɨɧɟɧɢɹ ɝɟɧɟɪɚɥɶɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ [.

3)ɇɚɣɬɢ ɧɚɢɦɟɧɶɲɢɣ ɢ ɧɚɢɛɨɥɶɲɢɣ ɷɥɟɦɟɧɬɵ ɜɵɛɨɪɤɢ ɢ ɪɚɡ- ɦɚɯ ɜɵɛɨɪɤɢ D. Ɋɚɡɞɟɥɢɜ ɢɧɬɟɪɜɚɥ [min {; max {] ɧɚ 10

ɪɚɜɧɵɯ ɩɨ ɞɥɢɧɟ ɱɚɫɬɢɱɧɵɯ ɢɧɬɟɪɜɚɥɨɜ L1>. . . >L10, ɧɚɣɬɢ ɞɥɢɧɭ k ɷɬɢɯ ɢɧɬɟɪɜɚɥɨɜ.

3

4)ɇɚɣɬɢ ɱɚɫɬɨɬɵ z1>. . . >z10, ɝɞɟ zn ɤɨɥɢɱɟɫɬɜɨ ɷɥɟɦɟɧɬɨɜ ɜɵɛɨɪɤɢ, ɩɪɢɧɚɞɥɟɠɚɳɢɯ ɱɚɫɬɢɱɧɨɦɭ ɢɧɬɟɪɜɚɥɭ Ln.

5)ɉɨɫɬɪɨɢɬɶ ɝɢɫɬɨɝɪɚɦɦɭ ɱɚɫɬɨɬ.

2.ɉɪɢɦɟɪ ɜɵɩɨɥɧɟɧɢɹ ɪɚɛɨɬɵ 1

1.ɉɭɫɬɶ q = 81, = 6 ɢ = 2. ɉɪɢɦɟɧɹɹ ɤɨɦɚɧɞɭ Vwdwlvwlfv + Udqgrp Qxpehuv, ɨɬɤɪɨɟɦ ɞɢɚɥɨɝɨɜɨɟ ɨɤɧɨ, ɜ ɤɨɬɨɪɨɦ ɜɵɛɟɪɟɦ

Qrupdo ɜ ɩɨɥɟ Glvwulexwlrq, ɧɚɛɟɪɟɦ 81 ɜ ɩɨɥɟ Krz Pdq|?, 6 ɜ ɩɨɥɟ Phdq, 2 ɜ ɩɨɥɟ Vwdqgdug Ghyldwlrq ɢ ɡɚɤɪɨɟɦ ɨɤɧɨ (RN). ɉɨɥɭɱɚɟɦ ɜɵɛɨɪɤɭ ɨɛɴɟɦɨɦ q = 81. Ɂɚɤɥɸɱɢɦ ɩɨɥɭɱɟɧɧɵɟ ɱɢɫɥɚ ɜ ɫɤɨɛɤɢ, ɩɨɫɬɚɜɢɦ ɩɟɪɟɞ ɧɢɦɢ { = ɢ ɨ ɩ ɪ ɟ ɞ ɟ ɥ ɢ ɦ ɩɨɥɭɱɟɧɧɵɣ ɜɟɤɬɨɪ ɜ ɫɪɟɞɟ SWP, ɢɫɩɨɥɶɡɭɹ ɤɨɦɚɧɞɭ Ghilqlwlrqv + Qhz Ghilqlwlrq. ɂɬɚɤ, ɧɟɨɛɯɨɞɢɦɵɟ ɞɥɹ ɨɛɪɚɛɨɬɤɢ ɧɚɛɥɸɞɟɧɢɹ {l (l = 1>81) ɩɨɥɭɱɟɧɵ ɢ ɨɩɪɟɞɟɥɟɧɵ ɜ ɫɪɟɞɟ SWP.

{= (8=3517>4=8733>6=4708>3=1149>3=8416>5=9560>0=82944>

5=1135>3=9934>5=9443>9=0525>4=7898>6=3281>7=306>4=9179>10=27> 6=3688>4=7667>5=1026>7=6476>6=5043>6=3837>7=6560>2=5155>8=2462> 5=6789>2=8881>4=4386>4=9627>5=4835>2=9277>6=8404>4=9079>4=9531> 6=1121>9=0432>2=4218>3=1825>2=4474>4=1068>5=79>9=9930>3=4842> 5=8872>8=227>6=7383>5=9369>7=238>9=487 6>3=761 8>5=0165>5=0704> 6=9068>4=7252>4=1059>9=1947>6=9059>2=9462>6=7039>7=2671>5=3456> 4=1052>5=342 5>5=3026>5=8626>5=1571>5=7057>8=5192>5=8829>8=6556> 3=4126>8=8435>7=1837>8=6929>4=1479>2=3171>4=7185>2=3257>9=4463> 3=1624>4=3045).

2.ɂɫɩɨɥɶɡɭɹ ɤɨɦɚɧɞɵ Vwdwlvwlfv + Phdq, Vwdwlvwlfv + Y duldqfh

ɢVwdwlvwlfv + Vwdqgdug Ghyldwlrq, ɧɚɯɨɞɢɦ ɨɰɟɧɤɢ ɩɚɪɚɦɟɬɪɨɜ ɝɟɧɟɪɚɥɶɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ [, ɜɜɨɞɹ ɤɚɠɞɵɣ ɪɚɡ ɬɟɤɫɬɨɜɨɣ ɤɭɪɫɨɪ ɜ ɩɨɥɟ ɜɵɛɨɪɤɢ. ɉɨɥɭɱɚɟɦ: Mean(s): 5=6859, Variance(s): 4=4987, Standard deviation(s): 2=0848, ɬ.ɟ. {¯ = 5=6859, 2 v2 = 4=3462

4

3. ɇɚɣɞɟɦ ɧɚɢɦɟɧɶɲɢɣ ɢ ɧɚɢɛɨɥɶɲɢɣ ɷɥɟɦɟɧɬɵ ɜɵɛɨɪɤɢ ɢ ɪɚɡɦɚɯ ɜɵɛɨɪɤɢ D. ɉɪɢɦɟɧɹɹ ɤɨɦɚɧɞɭ Hydoxdwh, ɧɚɯɨɞɢɦ min { = 0=82944, max { = 10=27, D = max { min { = 9=4406. ɋ ɩɨɦɨɳɶɸ

Ghilqlwlrqv + Qhz Ghilqlwlrq ɨɩɪɟɞɟɥɢɦ ɜ ɫɪɟɞɟ SWP ɞɥɢɧɭ k

ɱɚɫɬɢɱɧɵɯ ɢɧɬɟɪɜɚɥɨɜ ɢ ɜɵɱɢɫɥɢɦ ɟɟ ɡɧɚɱɟɧɢɟ. ɉɨɥɭɱɚɟɦ

k = max { min { = 0=94406. 10

4. ɇɚɣɞɟɦ ɱɚɫɬɨɬɵ z1>. . . >z10, ɝɞɟ zn ɤɨɥɢɱɟɫɬɜɨ ɷɥɟɦɟɧɬɨɜ ɜɵɛɨɪɤɢ, ɩɪɢɧɚɞɥɟɠɚɳɢɯ ɱɚɫɬɢɱɧɨɦɭ ɢɧɬɟɪɜɚɥɭ Ln. Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ d ɥɟɜɵɣ ɤɨɧɟɰ ɢɧɬɟɪɜɚɥɚ [min {; max {] ɢ ɨɩɪɟɞɟɥɢɦ ɟɝɨ ɜ ɫɪɟɞɟ SWP: d = min {.

ɇɚɛɢɪɚɟɦ ɢ ɫ ɩɨɦɨɳɶɸ ɤɨɦɚɧɞɵ Hydoxdwh ɜɵɱɢɫɥɹɟɦ ɱɚɫɬɨɬɵ:

X81

z1 = Heaviside(d + k {n) = 1;

n=1 X81

z2 = (Heaviside(d+ 2k {n) Heaviside(d+ k {n)) = 5;

n=1 X81

z3 = (Heaviside(d+ 3k {n) Heaviside(d+ 2k {n) = 8;

n=1 X81

z4 = (Heaviside(d+ 4k {n) Heaviside(d+ 3k {n)) = 9;

n=1 X81

z5 = (Heaviside(d+5k {n) Heaviside(d+4k {n)) = 18;

n=1 X81

z6 = (Heaviside(d+6k {n) Heaviside(d+5k {n)) = 14;

n=1

5

X81

z7 = (Heaviside(d+7k {n) Heaviside(d+6k {n)) = 10;

n=1 X81

z8 = (Heaviside(d+ 8k {n) Heaviside(d+ 7k {n)) = 5;

n=1 X81

z9 = (Heaviside(d+ 9k {n) Heaviside(d+ 8k {n)) = 7;

n=1 X81

z10 = Heaviside(d + 10k + 0=1 {n)

n=1 X81

Heaviside(d + 9k {n) = 4=0.

n=1

ɋ ɩɨɦɨɳɶɸ ɩɢɤɬɨɝɪɚɦɦɵ Qhz Ghilqlwlrq ɨɩɪɟɞɟɥɢɦ ɩɨɥɭɱɟɧ- ɧɵɟ ɱɚɫɬɨɬɵ ɜ ɫɪɟɞɟ SWP. z1 = 1, z2 = 5, z3 = 8, z4 = 9,

z5 = 18, z6 = 14, z7 = 10, z8 = 5, z9 = 7, z10 = 4.

ɉɨɫɬɪɨɢɦ ɬɚɛɥɢɰɭ ɱɚɫɬɨɬ:

5. Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ ɝɢɫɬɨɝɪɚɦɦɵ ɱɚɫɬɨɬ ɧɚɛɢɪɚɟɦ ɫɥɟɞɭɸɳɟɟ ɞɥɢɧɧɨɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɜɵɪɚɠɟɧɢɟ (ɩɪɢ ɷɬɨɦ ɜ ɤɨɧɰɟ ɤɚɠɞɨɣ ɫɬɪɨɤɢ ɧɚɞɨ ɩɪɢɦɟɧɹɬɶ ɤɨɦɚɧɞɭ Lqvhuw + Vsdflqj + Euhdn + Doorzeuhdn, ɱɬɨɛɵ ɷɬɢ ɫɬɪɨɤɢ ɫɨɡɞɚɜɚɥɢ ɟɞɢɧɨɟ ɜɵɪɚɠɟɧɢɟ):

(d>0>d>z1>d+k>z1>d+k>0>d+k>z2>d+2k>z2>d+2k>0>d+2k>z3> d+ 3k>z3>d+ 3k>0>d+ 3k>z4>d+ 4k>z4>d+ 4k>0>d+ 4k>z5>d+ 5k>z5>d+5k>0>d+5k>z6>d+6k>z6>d+6k>0>d+6k>z7>d+7k>z7>

d+ 7k>0>d + 7k>z8>d + 8k>z8>d + 8k>0>d + 8k>z9>d + 9k>z9>

d+ 9k>0>d + 9k>z10>d + 10k>z10>d + 10k>0>d>0).

ɋɩɨɦɨɳɶɸ ɤɨɦɚɧɞɵ Sorw 2G + Uhfwdqjxodu ɫɬɪɨɢɦ ɝɪɚɮɢɤ.

ɂɫɩɨɥɶɡɭɹ ɩɢɤɬɨɝɪɚɦɦɭ Surshuwlhv, ɨɬɤɪɵɜɚɟɦ ɨɤɧɨ Sorw Surshuwlhv, ɜ ɪɚɡɞɟɥɟ Lwhpv Sorwwhg ɜɵɛɢɪɚɟɦ Phglxp ɤɚɤ Olqh

6

18

16

14

12

10

8

6

4

2

Ɋɢɫ. 1. Ƚɢɫɬɨɝɪɚɦɦɚ ɱɚɫɬɨɬ

Wklfnqhvv ɢ ɡɚɤɪɵɜɚɟɦ ɨɤɧɨ (RN). ɉɨɥɭɱɚɟɦ ɝɢɫɬɨɝɪɚɦɦɭ ɱɚɫɬɨɬ

(ɪɢɫ. 1).

3.Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ 2. Ⱦɨɜɟɪɢɬɟɥɶɧɵɟ ɢɧɬɟɪɜɚɥɵ

1)ɋɨɫɬɚɜɢɬɶ ɜɵɛɨɪɤɭ ɨɛɴɟɦɨɦ q = 10 + P (P = Q mod 6 + Q mod 5, Q ɧɨɦɟɪ ɫɬɭɞɟɧɬɚɩɨ ɫɩɢɫɤɭ ɝɪɭɩɩɵ) ɢɡɧɨɪɦɚɥɶ- ɧɨɣ ɝɟɧɟɪɚɥɶɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ [ ɫ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɨɠɢɞɚ- ɧɢɟɦ = P ɢ ɫɪɟɞɧɢɦ ɤɜɚɞɪɚɬɢɱɟɫɤɢɦ ɨɬɤɥɨɧɟɧɢɟɦ = 3.

2)ɇɚɣɬɢ ɨɰɟɧɤɭ {¯ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨɨɠɢɞɚɧɢɹ ɢ ɢɫɩɪɚɜɥɟɧɧɵɟ ɨɰɟɧɤɢ v2 ɢ v ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɞɢɫɩɟɪɫɢɢ 2 ɢɫɪɟɞɧɟɝɨɤɜɚɞɪɚ- ɬɢɱɟɫɤɨɝɨ ɨɬɤɥɨɧɟɧɢɹ ɝɟɧɟɪɚɥɶɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ [.

3)ɋ ɞɨɜɟɪɢɬɟɥɶɧɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ = 0=95 ɧɚɣɬɢ ɞɨɜɟɪɢɬɟɥɶ-

7

ɧɵɣ ɢɧɬɟɪɜɚɥ ɞɥɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ .

4)ɋ ɞɨɜɟɪɢɬɟɥɶɧɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ = 0=95 ɧɚɣɬɢ ɞɨɜɟɪɢɬɟɥɶ- ɧɵɣ ɢɧɬɟɪɜɚɥ ɞɥɹ ɞɢɫɩɟɪɫɢɢ 2.

5)ɂɫɩɨɥɶɡɭɹ ɩɨɥɭɱɟɧɧɵɣ ɞɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ ɞɥɹ ɞɢɫɩɟɪ- ɫɢɢ 2, ɧɚɣɬɢ ɩɪɢɛɥɢɠɟɧɧɵɣ ɞɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ ɞɥɹ ɫɪɟɞɧɟɝɨ ɤɜɚɞɪɚɬɢɱɟɫɤɨɝɨ ɨɬɤɥɨɧɟɧɢɹ .

4.ɉɪɢɦɟɪ ɜɵɩɨɥɧɟɧɢɹ ɪɚɛɨɬɵ 2

1.ɉɭɫɬɶ q = 10, = 6 ɢ = 2. ɉɪɢɦɟɧɹɹ ɤɨɦɚɧɞɭ Vwdwlvwlfv + Udqgrp Qxpehuv, ɨɬɤɪɨɟɦ ɞɢɚɥɨɝɨɜɨɟ ɨɤɧɨ, ɜ ɤɨɬɨɪɨɦ ɜɵɛɟɪɟɦ

Qrupdo ɜ ɩɨɥɟ Glvwulexwlrq, ɧɚɛɟɪɟɦ 10 ɜ ɩɨɥɟ Krz Pdq|?, 6 ɜ ɩɨɥɟ Phdq, 2 ɜ ɩɨɥɟ Vwdqgdug Ghyldwlrq ɢ ɡɚɤɪɨɟɦ ɨɤɧɨ (RN).

ɉɨɥɭɱɚɟɦ ɜɵɛɨɪɤɭ ɨɛɴɟɦɨɦ q = 10: 3=9934, 5=9443, 9=0525, 4=7898,

6=3281, 7=306, 4=9179, 10=27, 6=368 8, 4=7667.

2.ɂɫɩɨɥɶɡɭɹ ɤɨɦɚɧɞɵ Vwdwlvwlfv + Phdq, Vwdwlvwlfv + Y duldqfh

ɢVwdwlvwlfv + Vwdqgdug Ghyldwlrq, ɧɚɯɨɞɢɦ ɨɰɟɧɤɢ ɩɚɪɚɦɟɬɪɨɜ ɝɟɧɟɪɚɥɶɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ [, ɜɜɨɞɹ ɤɚɠɞɵɣ ɪɚɡ ɬɟɤɫɬɨɜɨɣ ɤɭɪɫɨɪ ɜ ɩɨɥɟ ɜɵɛɨɪɤɢ. ɉɨɥɭɱɚɟɦ: Mean(s): 6=3738, Variance(s): 4=0321, Standard deviation(s): 2=008, ɬ.ɟ. {¯ = 6=3738, 2 v2 = 4=0321

ɢv = 2=008. ɇɚɩɨɦɧɢɦ, ɱɬɨ

ɂɫɩɨɥɶɡɭɹ ɤɨɦɚɧɞɭ Ghilqlwlrqv + Qhz Ghilqlwlrq, ɨɩɪɟɞɟɥɢɦ

ɜɫɪɟɞɟ SWP {¯ = 6=3738 ɢ v = 2=008.

3.ɂɡɜɟɫɬɧɨ, ɱɬɨ ɞɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ ɞɥɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɠɢɞɚɧɢɹ ɧɨɪɦɚɥɶɧɨɣ ɝɟɧɟɪɚɥɶɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ [ ɫ ɧɟɢɡɜɟɫɬ-

ɧɨɣ ɞɢɫɩɟɪɫɢɟɣ 2 ɧɚɯɨɞɢɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:

· ¸

5 [d>e] = {¯ wsv >{¯ + wsv , q q

8

ɝɞɟ w ɨɩɪɟɞɟɥɹɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ W, ɪɚɫɩɪɟɞɟɥɟɧ- ɧɨɣ ɩɨ ɡɚɤɨɧɭ ɋɬɶɸɞɟɧɬɚ ɫ = q 1 ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ, ɬɚɤ, ɱɬɨ

Pr( w ? W ? w) = , ɬ.ɟ. Pr(W ? w) = + (1 )@2 = (1 + )@2.

ɂɫɩɨɥɶɡɭɹ ɮɭɧɤɰɢɢ SWP, ɩɨɥɭɱɚɟɦ w = TInv((1 + )@2>q 1).

=[4=9374>7=8102] .

4.ɂɡɜɟɫɬɧɨ, ɱɬɨ ɞɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ ɞɥɹ ɞɢɫɩɟɪɫɢɢ 2 ɧɨɪɦɚɥɶɧɨɣ ɝɟɧɟɪɚɥɶɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ [ ɧɚɯɨɞɢɬɫɹ ɫɥɟɞɭɸɳɢɦ

ɝɞɟ Y1 ɢ Y2 ɨɩɪɟɞɟɥɹɸɬɫɹ ɫ ɩɨɦɨɳɶɸ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ "2, ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɩɨ ɡɚɤɨɧɭ ɉɢɪɫɨɧɚ ɫ = q 1 ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ,

ɬɚɤ, ɱɬɨ Pr(W ? Y1) = (1 )@2 ɢ Pr(W ? Y2) = + (1 )@2 = (1 + )@2.

ɂɫɩɨɥɶɡɭɹ ɮɭɧɤɰɢɢ SWP, ɩɨɥɭɱɚɟɦ Y1 = ChiSquareInv((1

)@2>q 1) ɢ Y2 = ChiSquareInv((1 + )@2>q 1).

ɋ ɩɨɦɨɳɶɸ ɤɨɦɚɧɞɵ Hydoxdwh Qxphulfdoo| ɩɨɥɭɱɚɟɦ

= [1=9076>13=438] .

5. ɋ ɩɨɦɨɳɶɸ ɤɨɦɚɧɞɵ Hydoxdwh, ɢɫɩɨɥɶɡɭɹ ɩɨɥɭɱɟɧɧɵɣ ɞɨɜɟ- ɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ ɞɥɹ ɞɢɫɩɟɪɫɢɢ 2, ɧɚɯɨɞɢɦ ɩɪɢɛɥɢɠɟɧɧɵɣ ɞɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ ɞɥɹ ɫɪɟɞɧɟɝɨ ɤɜɚɞɪɚɬɢɱɟɫɤɨɝɨ ɨɬɤɥɨɧɟɧɢɹ

= [1=3812>3=6658] .

9

5.Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ 3. Ʉɪɢɬɟɪɢɣ ɫɨɝɥɚɫɢɹ ɉɢɪɫɨɧɚ

1)ɋɨɫɬɚɜɢɬɶ ɜɵɛɨɪɤɭ ɨɛɴɟɦɨɦ q = 81 + P (P = Q mod 9 + Q mod 7, Q ɧɨɦɟɪ ɫɬɭɞɟɧɬɚ ɩɨ ɫɩɢɫɤɭ ɝɪɭɩɩɵ) ɢɡ ɧɨɪɦɚɥɶ- ɧɨɣ ɝɟɧɟɪɚɥɶɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ [ ɫ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɨɠɢɞɚ- ɧɢɟɦ = P ɢ ɫɪɟɞɧɢɦ ɤɜɚɞɪɚɬɢɱɟɫɤɢɦ ɨɬɤɥɨɧɟɧɢɟɦ = 3.

2)ɇɚɣɬɢ ɧɚɢɦɟɧɶɲɢɣ ɢ ɧɚɢɛɨɥɶɲɢɣ ɷɥɟɦɟɧɬɵ ɜɵɛɨɪɤɢ ɢ ɟɟ ɚɦɩɥɢɬɭɞɭ D. Ɋɚɡɞɟɥɢɜ ɢɧɬɟɪɜɚɥ [min {; max {] ɧɚ 10 ɪɚɜɧɵɯ

ɩɨ ɞɥɢɧɟ ɱɚɫɬɢɱɧɵɯ ɢɧɬɟɪɜɚɥɨɜ L1>. . . >L10, ɧɚɣɬɢ ɞɥɢɧɭ k ɷɬɢɯ ɢɧɬɟɪɜɚɥɨɜ.

3)ɇɚɣɬɢ ɱɚɫɬɨɬɵ z1>. . . >z10, ɝɞɟ zn ɤɨɥɢɱɟɫɬɜɨ ɷɥɟɦɟɧɬɨɜ ɜɵɛɨɪɤɢ, ɩɪɢɧɚɞɥɟɠɚɳɢɯ ɱɚɫɬɢɱɧɨɦɭ ɢɧɬɟɪɜɚɥɭ Ln.

4)ɋ ɩɨɦɨɳɶɸ ɤɪɢɬɟɪɢɹ ɫɨɝɥɚɫɢɹ ɉɢɪɫɨɧɚ ɩɪɢ ɭɪɨɜɧɟ ɡɧɚɱɢ- ɦɨɫɬɢ = 0=05 ɩɪɨɜɟɪɢɬɶ ɝɢɩɨɬɟɡɭ ɨ ɬɨɦ, ɱɬɨ ɝɟɧɟɪɚɥɶɧɚɹ ɫɨɜɨɤɭɩɧɨɫɬɶ ɪɚɫɩɪɟɞɟɥɟɧɚ ɩɨ ɧɨɪɦɚɥɶɧɨɦɭ ɡɚɤɨɧɭ.

6.ɉɪɢɦɟɪ ɜɵɩɨɥɧɟɧɢɹ ɪɚɛɨɬɵ 3

1.ɉɭɫɬɶ q = 81, = 6 ɢ = 2. Ⱥɧɚɥɨɝɢɱɧɨ ɪɚɡɞɟɥɭ 1 ɩɪɢɦɟɪɚ ɪɚɛɨɬɵ 1 ɩɨɫɬɪɨɢɦ ɧɭɠɧɭɸ ɞɥɹ ɨɛɪɚɛɨɬɤɢ ɜɵɛɨɪɤɭ ɨɛɴɟɦɨɦ q =

81.Ⱥɧɚɥɨɝɢɱɧɨ ɪɚɡɞɟɥɭ 2 ɩɪɢɦɟɪɚ ɪɚɛɨɬɵ 1 ɧɚɯɨɞɢɦ ɨɰɟɧɤɢ ɩɚɪɚ- ɦɟɬɪɨɜ ɢ ɝɟɧɟɪɚɥɶɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ [. ɉɨɥɭɱɚɟɦ: Mean(s):

5=6859, Standard deviation(s): 2=0848, ɬ.ɟ. {¯ = 5=6859 ɢ v =

2=0848.

2. ɗɬɨɬ ɪɚɡɞɟɥ ɫɨɜɩɚɞɚɟɬ ɫ ɪɚɡɞɟɥɨɦ 3 ɩɪɢɦɟɪɚ ɪɚɛɨɬɵ 1. ɉɨɥɭ-

+ Qhz Ghilqlwlrq ɨɩɪɟɞɟɥɢɦ ɜ ɫɪɟɞɟ SWP ɞɥɢɧɭ k ɱɚɫɬɢɱɧɵɯ

10