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llist:=[13,sin(x),`ɭɦɧɵɣ ɫɬɭɞɟɧɬ`];
llist >VLQ[ɭɦɧɵɣɫɬɭɞɟɧɬ@
a:=op(llist);
ɚ VLQ[ɭɦɧɵɣɫɬɭɞɟɧɬ
Ʉɫɩɢɫɤɭɧɨɜɵɟɷɥɟɦɟɧɬɵɩɪɢɫɨɟɞɢɧɹɸɬɫɹɬɚɤ
llist:=[a,`ɧɟɝɥɭɩɵɣ ɫɬɭɞɟɧɬ`, cos(Pi)];
llist >VLQ[ɭɦɧɵɣɫɬɭɞɟɧɬɧɟɝɥɭɩɵɣɫɬɭɞɟɧɬ@
ȿɫɥɢ ɚɥɝɨɪɢɬɦ ɡɚɥɨɠɟɧɧɵɣ ɜ ɩɪɨɰɟɞɭɪɟ DartDensity ɬɟɩɟɪɶ ɜɚɦ
ɩɨɧɹɬɟɧ ɧɚɩɢɲɢɬɟ ɩɪɨɰɟɞɭɪɭ
ɪɨɣɦɨɠɧɨɨɩɪɟɞɟɥɢɬɶɩɥɨɬɧɨɫɬɶ
ɦɨɳɶɸɧɚɣɞɟɧɧɨɣɮɭɧɤɰɢɢ
BallDensity(n,k) ɫ ɩɨɦɨɳɶɸ ɤɨɬɨ
gxɪɚɫɩɪɟɞɟɥɟɧɢɹɫɜɏɁɚɬɟɦɫɩɨ
gxɜɵɜɟɞɢɬɟɮɨɪɦɭɥɭPa X b b ±a
ǮȇȋȇȔȏȌ 3
ɉɭɫɬɶXɢY± ɧɟɩɪɟɪɵɜɧɵɟ ɫɥɭɱɚɣɧɵɟɜɟɥɢɱɢɧɵɤɚɠɞɚɹ ɢɡ ɤɨɬɨɪɵɯ
ɪɚɜɧɨɦɟɪɧɨɪɚɫɩɪɟɞɟɥɟɧɚɧɚɨɬɪɟɡɤɟ>@
Z XY
ɇɚɩɢɲɢɬɟɩɪɨɰɟɞɭɪɭSum_X_and_Y(n,k)ɫɩɨɦɨɳɶɸɤɨɬɨɪɨɣɦɨɠɧɨ
ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɥɨɬɧɨɫɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
hzɫɜZ
ɉɨ ɪɟɡɭɥɶɬɚɬɚɦ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɧɚɣɞɢɬɟ ɬɨɱɧɨɟ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɜɵ
ɪɚɠɟɧɢɟ ɞɥɹ
hz ɫ ɭɱɟɬɨɦ ɫɥɟɞɭɸɳɟɣ ɞɨɩɨɥɧɢɬɟɥɶɧɨɣ ɢɧɮɨɪɦɚɰɢɢ
hzɹɜɥɹɟɬɫɹɤɭɫɨɱɧɨɥɢɧɟɣɧɨɣɮɭɧɤɰɢɟɣ
Ⱦɭɷɥɢ ɜ ɩɨɫɟɥɤɟ ɋɦɨɥɹɧɵ ɪɟɞɤɨ ɡɚɤɚɧɱɢɜɚɸɬɫɹ ɩɟɱɚɥɶɧɵɦ ɢɫɯɨɞɨɦ
Ⱦɟɥɨɜɬɨɦɱɬɨɤɚɠɞɵɣɞɭɷɥɹɧɬɩɪɢɛɵɜɚɟɬɧɚɦɟɫɬɨɜɫɬɪɟɱɢɜɫɥɭɱɚɣɧɵɣɦɨ
ɦɟɧɬɜɪɟɦɟɧɢɦɟɠɞɭɢɱɚɫɚɦɢɭɬɪɚɢɩɪɨɠɞɚɜɫɨɩɟɪɧɢɤɚɦɢɧɭɬɭɞɚɥɹɟɬ
ɫɹȼɫɥɭɱɚɟɠɟɩɪɢɛɵɬɢɹɩɨɫɥɟɞɧɟɝɨɜɷɬɢɩɹɬɶɦɢɧɭɬɞɭɷɥɶɫɨɫɬɨɢɬɫɹɄɚɤɨɜɚ
ɜɟɪɨɹɬɧɨɫɬɶɬɨɝɨɱɬɨɞɭɷɥɶɡɚɤɚɧɱɢɜɚɟɬɫɹɩɨɟɞɢɧɤɨɦ"
ȿɫɥɢ ɜɪɟɦɟɧɚ ɩɪɢɛɵɬɢɹ ɞɭɷɥɹɧɬɨɜ ɹɜɥɹɸɬɫɹ ɫɥɭɱɚɣɧɵɦɢ ɜɟɥɢɱɢɧɚɦɢ
X
ɬɨɜɜɨɩɪɨɫɟɡɚɞɚɱɢɪɟɱɶɢɞɟɬɨɫɥɭɱɚɣɧɨɣɜɟɥɢɱɢɧɟY _X±X_ɜɵɪɚɠɚɸ
ɢ
X
ɳɟɣɞɥɢɧɭɜɪɟɦɟɧɧɨɝɨɢɧɬɟɪɜɚɥɚɦɟɠɞɭɩɨɹɜɥɟɧɢɹɦɢɞɜɭɯɞɭɷɥɹɧɬɨɜɉɨɷɬɨɦɭ
ɢɫɤɨɦɚɹ ɜɟɪɨɹɬɧɨɫɬɶ ɛɭɞɟɬ ɧɚɣɞɟɧɚ ɟɫɥɢ ɛɭɞɟɬ ɧɚɣɞɟɧɚ ɧɚɤɨɩɢɬɟɥɶɧɚɹ ɮɭɧɤ
j
j
ɰɢɹ
FyɫɥɭɱɚɣɧɨɣɜɟɥɢɱɢɧɵYɩɨɫɤɨɥɶɤɭ
§·§·
PY F
d
¨¸¨¸
©¹©¹
ɇɚɩɢɲɢɬɟ ɩɪɨɰɟɞɭɪɭ Meeting(n,k) ɫ ɩɨɦɨɳɶɸ ɤɨɬɨɪɨɣ ɦɨɠɧɨ
ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɥɨɬɧɨɫɬɶ
hy ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫ ɜ Y ɉɨ
ɪɟɡɭɥɶɬɚɬɚɦ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɧɚɣɞɢɬɟ ɬɨɱɧɨɟ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɜɵɪɚɠɟ
ɧɢɟɞɥɹ
ɱɟɧɢɟɷɬɨɣɮɭɧɤɰɢɢɜɬɨɱɤɟ
hyɇɚɣɞɢɬɟɧɚɤɨɩɢɬɟɥɶɧɭɸɮɭɧɤɰɢɸFyɫɜYɢɡɚɬɟɦɡɧɚ
ǺȘȒȕȉȔȇȦȉȌȗȕȦșȔȕȘșȣȏțȕȗȓȚȒȇǨȇȌȐȘȇ
ɉɭɫɬɶɞɚɧɚɩɨɥɧɚɹɝɪɭɩɩɚɫɨɛɵɬɢɣHH«Hnɬɟɷɬɢɫɨɛɵɬɢɹɩɨɩɚɪɧɨ
ɧɟɫɨɜɦɟɫɬɧɵɚɢɯɨɛɴɟɞɢɧɟɧɢɟɟɫɬɶɜɫɟɩɪɨɫɬɪɚɧɫɬɜɨɢɫɯɨɞɨɜ
ɧɚɡɨɜɟɦɝɢɩɨɬɟɡɚɦɢɉɭɫɬɶɬɚɤɠɟɢɦɟɸɬɫɹɫɨɛɵɬɢɹ
E
E«Emɤɨɬɨɪɵɟɦɨɝɭɬ
ɩɨɜɥɢɹɬɶ ɧɚ ɜɟɪɨɹɬɧɨɫɬɢ ɝɢɩɨɬɟɡ ɇɚɡɨɜɟɦ ɷɬɢ ɫɨɛɵɬɢɹ ɭɥɢɤɚɦɢ ɉɪɟɞɜɚɪɢ
PH
ɬɟɥɶɧɨɢɡɜɟɫɬɧɵɜɟɪɨɹɬɧɨɫɬɢ
ɥɹɟɬɧɚɯɨɞɢɬɶɧɨɜɭɸ ɭɫɥɨɜɧɭɸɜɟɪɨɹɬɧɨɫɬɶ
ɧɚɥɢɱɢɹɭɥɢɤɢ
E
ɝɢɩɨɬɟɡHii «nɎɨɪɦɭɥɚȻɚɣɟɫɚɩɨɡɜɨ
i
PH
ɝɢɩɨɬɟɡɵ Hiɩɪɢɭɫɥɨɜɢɢ
i_Ej
:ɗɬɢɫɨɛɵɬɢɹ
_
PH E
ij
PE H PH
n
PE H PH
¦
k
ji i
_
kk
i «nj «m
_
ǮȇȋȇȔȏȌ 4ǫȒȦȕȖȗȌȋȌȒȌȔȏȦȕȋȔȕȊȕȏȎșȗȌȜȉȏȋȕȉȈȕȒȌȎȔȌȐbbbȈȚȋȚșȖȗȕȉȕȋȏș
ȘȦȋȉȇșȌȘșȇȘȏȘȜȕȋȇȓȏȏ²ȑȇȍȋȢȐșȌȉȕȎȓȕȍȔȢȏȘȜȕȋȇ²²²²ǷȇȔȌȌ
ȞȌȒȕȉȌȑȑȇȍȋȢȐȏȎȑȕșȕȗȢȜȏȓȌȒȗȕȉȔȕ ȕȋȔȚ ȏȎ ȈȕȒȌȎȔȌȐ bbbȖȕȋȉȌȗȊ
ȒȏȘȣșȌȘșȏȗȕȉȇȔȏȥǷȌȎȚȒȣșȇșȢșȌȘșȏȗȕȉȇȔȏȦȘȉȌȋȌȔȢȉȘȒȌȋȚȥȠȚȥșȇȈȒȏȝȚ
ȈȕȒȌȎȔȏ ȞȏȘȒȕȈȕȒȣȔȢȜ ² ² ²²
b
b
b
Ɉɬɤɪɨɣɬɟ ɮɚɣɥ ȻɚɣɟɫɚɎɨɪɦɭɥɚPZV ɢ ɜɚɦ ɫɬɚɧɟɬ ɞɨɫɬɭɩɧɚ ɩɪɨɰɟɞɭɪɚ
BayesFormula(prior,conditional ɫɩɢɫɨɤ SULRU ɞɨɥɠɟɧ ɫɨɞɟɪɠɚɬɶ ɢɫ
ɯɨɞɧɵɟ ɜɟɪɨɹɬɧɨɫɬɢ ɝɢɩɨɬɟɡ ɫɩɢɫɨɤ FRQGLWLRQDO ɞɨɥɠɟɧ ɫɨɞɟɪɠɚɬɶɦɚɬɪɢɰɭ
ɪɚɡɦɟɪɚnumɧɚɩɨɡɢɰɢɢijɤɨɬɨɪɨɣɧɚɯɨɞɢɬɫɹɭɫɥɨɜɧɚɹɜɟɪɨɹɬɧɨɫɬɶPE
j_Hi
ȼ 0DSOH ɷɥɟɦɟɧɬɵ ɧɟɤɨɬɨɪɨɣ ɦɚɬɪɢɰɵ
ɫɩɢɫɤɚ ɢɡ
ɦɟɧɬɨɜ
M
m ɷɥɟɦɟɧɬɨɜ ɩɪɢ ɷɬɨɦ iɦ ɷɥɟɦɟɧɬɨɦ ɫɩɢɫɤɚ ɹɜɥɹɟɬɫɹ ɫɩɢɫɨɤ ɷɥɟ
iɣ ɫɬɪɨɤɢ ɦɚɬɪɢɰɵ Ɇ ɇɚɩɪɢɦɟɪ ɜɜɨɞ ɷɥɟɦɟɧɬɨɜ ɦɚɬɪɢɰɵ
§·
¨¸
©¹
ɨɫɭɳɟɫɬɜɥɹɟɬɫɹɬɚɤ
Ɇɪɚɡɦɟɪɚmun ɜɜɨɞɹɬɫɹ ɜ ɜɢɞɟ
[[13,2,-4],[2.7,1,0]];
ɋɩɨɦɨɳɶɸɩɪɨɰɟɞɭɪɵBayesFormulaɪɟɲɢɬɟɫɥɟɞɭɸɳɢɟɡɚɞɚɱɢ
ɇɚɣɞɢɬɟ ɜɟɪɨɹɬɧɨɫɬɢ Pb
ɫɬɜɢɹɤɚɠɞɨɝɨɢɡɡɚɛɨɥɟɜɚɧɢɣb
_ Pbi_ ± Pbi_± Pbi_± ± ɩɪɢɫɭɬ
i
i ɩɪɢɭɫɥɨɜɢɢɫɨɨɬɜɟɬɫɬɜɭɸ
i
ɳɟɝɨɪɟɡɭɥɶɬɚɬɚɬɟɫɬɢɪɨɜɚɧɢɹ±±±±
ȼɟɪɨɹɬɧɨɫɬɶ ɢɡɝɨɬɨɜɥɟɧɢɹ ɢɡɞɟɥɢɹ ɫ ɛɪɚɤɨɦ ɧɚ ɞɚɧɧɨɦ ɩɪɟɞɩɪɢɹɬɢɢ
ɪɚɜɧɚ ɉɟɪɟɞ ɜɵɩɭɫɤɨɦ ɢɡɞɟɥɢɟ ɩɨɞɜɟɪɝɚɟɬɫɹ ɭɩɪɨɳɟɧɧɨɣ ɩɪɨɜɟɪɤɟ
ɤɨɬɨɪɚɹ ɩɪɨɩɭɫɤɚɟɬ ɛɟɡɞɟɮɟɤɬɧɨɟ ɢɡɞɟɥɢɟ ɫ ɜɟɪɨɹɬɧɨɫɬɶɸ ɚ ɢɡɞɟɥɢɟ
ɫɞɟɮɟɤɬɨɦ±ɫɜɟɪɨɹɬɧɨɫɬɶɸ
ɇɚɣɞɢɬɟɚɜɟɪɨɹɬɧɨɫɬɶɬɨɝɨɱɬɨɢɡɞɟɥɢɟɜɵɞɟɪɠɚɜɲɟɟɭɩɪɨɳɟɧɧɭɸ
ɩɪɨɜɟɪɤɭ ɨɤɚɠɟɬɫɹ ɛɪɚɤɨɜɚɧɧɵɦ ɛ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ ɱɬɨ ɢɡɞɟɥɢɟ
ɧɟɜɵɞɟɪɠɚɜɲɟɟɭɩɪɨɳɟɧɧɭɸɩɪɨɜɟɪɤɭɨɤɚɠɟɬɫɹɛɟɡɞɟɮɟɤɬɧɵɦ
ɋɬɪɚɯɨɜɚɹ ɤɨɦɩɚɧɢɹ ɪɚɡɞɟɥɹɟɬ ɡɚɫɬɪɚɯɨɜɚɧɧɵɯɩɨ ɝɪɭɩɩɚɦ ɪɢɫɤɚ ɫɥɟ
ɞɭɸɳɢɦɨɛɪɚɡɨɦɩɟɪɜɚɹɝɪɭɩɩɚ±ɦɚɥɵɣɪɢɫɤɜɬɨɪɚɹɝɪɭɩɩɚ±ɫɪɟɞɧɢɣɪɢɫɤ
ɬɪɟɬɶɹ ɝɪɭɩɩɚ ± ɛɨɥɶɲɨɣ ɪɢɫɤ ɋɪɟɞɢ ɤɥɢɟɧɬɨɜ ± ɩɟɪɜɨɣ ɝɪɭɩɩɵ ɪɢɫɤɚ
±ɜɬɨɪɨɣɝɪɭɩɩɵɪɢɫɤɚ ±ɬɪɟɬɶɟɣɝɪɭɩɩɵɪɢɫɤɚɋɬɚɬɢɫɬɢɤɚɪɚɛɨɬɵ
ɤɨɦɩɚɧɢɢɩɨɤɚɡɚɥɚɱɬɨɧɟɨɛɯɨɞɢɦɨɫɬɶɜɵɩɥɚɱɢɜɚɬɶɫɬɪɚɯɨɜɨɟɜɨɡɧɚɝɪɚɠɞɟɧɢɟ
ɡɚɫɬɪɚɯɨɜɚɧɧɵɦɢɡɩɟɪɜɨɣɝɪɭɩɩɵɧɚɫɬɭɩɚɟɬɫɜɟɪɨɹɬɧɨɫɬɶɸɢɡɜɬɨɪɨɣ±
ɫɜɟɪɨɹɬɧɨɫɬɶɸɢɡɬɪɟɬɶɟɣ±ɫɜɟɪɨɹɬɧɨɫɬɶɸ
x
x
B
ɇɚɣɞɢɬɟɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨɱɬɨɩɨɥɭɱɢɜɲɢɣɜɨɡɧɚɝɪɚɠɞɟɧɢɟɡɚɫɬɪɚɯɨ
ɜɚɧɧɵɣɨɬɧɨɫɢɬɫɹɤɝɪɭɩɩɟɦɚɥɨɝɨɪɢɫɤɚ
ǺȘȒȕȉȔȢȌȖȒȕșȔȕȘșȏȔȌȖȗȌȗȢȉȔȢȜ
ȘȒȚȞȇȐȔȢȜȉȌȒȏȞȏȔ
ǮȇȋȇȔȏȌ 5
ɉɭɫɬɶfx ± ɩɥɨɬɧɨɫɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɧɟɩɪɟɪɵɜɧɨɣ ɫ ɜ ɏ Ɍɨɝɞɚ
ɭɫɥɨɜɧɨɣɩɥɨɬɧɨɫɬɶɸ ɩɪɢ ɭɫɥɨɜɢɢɫɜɟɪɲɟɧɢɹ ɫɨɛɵɬɢɹ
ɪɨɹɬɧɨɫɬɶɸ
PEɧɚɡɵɜɚɟɬɫɹɮɭɧɤɰɢɹ
°
_
fxE
®
°
¯
fx
ɟɫɥɢ
PE
ɟɫɥɢ
ɚɜɵɪɚɠɟɧɢɟ
ȿɫ ɩɨɥɨɠɢɬɟɥɶɧɨɣ ɜɟ
E
E
PAE f x Edx
_ _
³
xA
ɧɚɡɵɜɚɟɬɫɹɭɫɥɨɜɧɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ ɫɨɛɵɬɢɹȺɩɪɢɭɫɥɨɜɢɢɱɬɨɩɪɨɢɡɨɲɥɨ ɫɨ
ɛɵɬɢɟȿ
Ⱦɨɤɚɠɢɬɟɱɬɨ
_
PAE
PAE
PE
ɋɤɚɠɟɦɱɬɨɧɟɩɪɟɪɵɜɧɚɹɫɜXɢɦɟɟɬ ɛɟɬɚɪɚɫɩɪɟɞɟɥɟɧɢɟɫɩɚɪɚɦɟɬ
ɪɚɦɢ ɚɢb ɟɫɥɢ ɩɥɨɬɧɨɫɬɶ ɟɟ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɡɚɞɚɟɬɫɹ ɛɟɬɚɮɭɧɤɰɢɟɣ ɫ ɩɚɪɚ
ɦɟɬɪɚɦɢɚɢb
°
Babx
ab
ɝɞɟ
ab x x dx
³
Bab
®
°
ɜɩɪɨɬɢɜɧɨɦɫɥɭɱɚɟ
¯
ab
ɟɫɥɢ
dd
xx x
ȼɱɚɫɬɧɨɫɬɢ
Bab
ab
ab
ȼɢɞɛɟɬɚɮɭɧɤɰɢɢɫɭɳɟɫɬɜɟɧɧɨ ɢɡɦɟɧɹɟɬɫɹ ɫ ɢɡɦɟɧɟɧɢɟɦ ɡɧɚɱɟɧɢɣɩɚɪɚɦɟɬ
ɪɨɜɚɢbɉɨɷɬɨɦɭ ɜɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɛɟɬɚɮɭɧɤɰɢɹɩɨɡɜɨɥɹɟɬɞɨɫɬɚɬɨɱɧɨɬɨɱɧɨ
ɨɰɟɧɢɬɶɧɟɢɡɜɟɫɬɧɵɣɡɚɤɨɧɪɚɫɩɪɟɞɟɥɟɧɢɹɧɟɩɪɟɪɵɜɧɨɣɫɥɭɱɚɣɧɨɣɜɟɥɢɱɢɧɵ
ɉɨɫɬɪɨɣɬɟɝɪɚɮɢɤɢɛɟɬɚɮɭɧɤɰɢɢɩɪɢɪɚɡɥɢɱɧɵɯɡɧɚɱɟɧɢɹɯɩɚɪɚɦɟɬ
ɪɨɜɚɢbɧɚɣɞɢɬɟɬɚɤɢɟɡɧɚɱɟɧɢɹɩɚɪɚɦɟɬɪɨɜɚɢbɩɪɢɤɨɬɨɪɵɯɛɟɬɚ
ɪɚɫɩɪɟɞɟɥɟɧɢɟɫɬɚɧɨɜɢɬɫɹ ɪɚɜɧɨɦɟɪɧɵɦ ɧɚɣɞɢɬɟɬɚɤɢɟ ɡɧɚɱɟɧɢɹ ɩɚ
ɪɚɦɟɬɪɨɜ ɚɢb ɩɪɢ ɤɨɬɨɪɵɯ ɛɟɬɚɪɚɫɩɪɟɞɟɥɟɧɢɟ ɞɨɫɬɢɝɚɟɬ ɦɚɤɫɢ
x
ɦɚɥɶɧɨɝɨɡɧɚɱɟɧɢɹɩɪɢ
ɉɪɢɦɟɪɩɨɫɬɪɨɟɧɢɹɝɪɚɮɢɤɚɮɭɧɤɰɢɢfx axcɩɪɢɡɚɞɚɧɧɵɯɡɧɚɱɟɧɢɹɯ
ɩɚɪɚɦɟɬɪɨɜac
restart:
f:=a*x^2+c: a:=2: c:=17: plot(f,x=-1..1);
ǷȏȘ
ɉɭɫɬɶɏ±ɫɥɭɱɚɣɧɚɹɜɟɥɢɱɢɧɚɜɵɪɚɠɚɸɳɚɹɜɟɪɨɹɬɧɨɫɬɶɷɮɮɟɤɬɢɜ
B
B
ɧɨɫɬɢɩɪɢɦɟɧɟɧɢɹɜɚɤɰɢɧɵɩɪɨɬɢɜɜɢɪɭɫɚɝɪɢɩɩɚ$+1ɇɚɨɫɧɨɜɚɧɢɢɷɤɫ
ɩɟɪɢɦɟɧɬɚɥɶɧɵɯɞɚɧɧɵɯɛɵɥɨɭɫɬɚɧɨɜɥɟɧɨɱɬɨɩɥɨɬɧɨɫɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɟfx
ɫɥɭɱɚɣɧɨɣɜɟɥɢɱɢɧɵ ɏɦɨɠɧɨɩɪɢɛɥɢɡɢɬɟɥɶɧɨɫɱɢɬɚɬɶɛɟɬɚɮɭɧɤɰɢɟɣɫɩɚɪɚ
ɦɟɬɪɚɦɢ ɚɢb. Ɉɞɧɚɤɨ ɞɚɥɶɧɟɣɲɢɟ ɢɫɩɵɬɚɧɢɹ ɫɩɨɫɨɛɧɵ ɭɬɨɱɧɢɬɶ ɢ ɡɧɚɱɢ
ɬɟɥɶɧɨɩɪɢɛɥɢɡɢɬɶ ɤɪɟɚɥɶɧɵɦ ɩɟɪɜɨɧɚɱɚɥɶɧɨɜɡɹɬɵɟ ɡɧɚɱɟɧɢɹɩɚɪɚɦɟɬɪɨɜ ɚ
ɢbɉɨɤɚɠɟɦɤɚɤɷɬɨɦɨɠɧɨɫɞɟɥɚɬɶɩɨɫɥɟnɞɨɩɨɥɧɢɬɟɥɶɧɵɯɢɫɩɵɬɚɧɢɣɜɚɤ
ɰɢɧɵ
ɉɭɫɬɶ Y ± ɞɢɫɤɪɟɬɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ ɜɵɪɚɠɚɸɳɚɹ ɱɢɫɥɨ ɭɫɩɟɲɧɵɯ
ɢɫɯɨɞɨɜ iɜnɢɫɩɵɬɚɧɢɹɯɋɩɨɦɨɳɶɸɧɟɫɥɨɠɧɵɯɜɵɤɥɚɞɨɤɦɨɠɧɨɩɨɥɭɱɢɬɶ
ɮɨɪɦɭɥɭɞɥɹɭɫɥɨɜɧɨɣɩɥɨɬɧɨɫɬɢfx_Y i
ai b j
_
fxY i
xx
D E
ij
ɡɞɟɫɶj n±i
Ⱦɪɭɝɢɦɢɫɥɨɜɚɦɢ ɩɪɢ ɧɚɥɢɱɢɢiɭɫɩɟɲɧɵɯɢɫɯɨɞɨɜɜ n ɢɫɩɵɬɚɧɢɹɯɧɨɜɚɹ
ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɷɮɮɟɤɬɢɜɧɨɫɬɢ ɜɚɤɰɢɧɵ ɬɚɤɠɟ ɛɭɞɟɬ ɛɟɬɚɮɭɧɤɰɢɟɣ
ɧɨɭɠɟɫɩɚɪɚɦɟɬɪɚɦɢDiɢEn ±i Ej
ȼɵɜɟɞɟɦ ɬɟɩɟɪɶ ɮɨɪɦɭɥɭ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ p
PC ɝɞɟ
n
ɋ±ɫɥɭɱɚɣɧɨɟɫɨɛɵɬɢɟɫɨɫɬɨɹɳɟɟɜɭɫɩɟɲɧɨɫɬɢɩɪɢɦɟɧɟɧɢɹɜɚɤɰɢɧɵɜɩɨɫɥɟ
ɞɭɸɳɟɦ±nɦɢɫɩɵɬɚɧɢɢɫɭɱɟɬɨɦɬɨɝɨɱɬɨɩɥɨɬɧɨɫɬɶɪɚɫɩɪɟɞɟɥɟɧɢɹ
fx
ɫ ɜ ɏ ɜɵɪɚɠɚɸɳɟɣ ɷɮɮɟɤɬɢɜɧɨɫɬɶ ɩɪɢɦɟɧɟɧɢɹ ɜɚɤɰɢɧɵ ɹɜɥɹɟɬɫɹ ɛɟɬɚ
ɮɭɧɤɰɢɟɣBaibjt
Ɉɛɨɡɧɚɱɢɦ ɫɥɭɱɚɣɧɨɟ ɫɨɛɵɬɢɟ Bt X t ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ ɫɥɭɱɚɣɧɨɣ
ɜɟɥɢɱɢɧɵ ɏ ɜɵɪɚɠɚɸɳɟɣ ɜɟɪɨɹɬɧɨɫɬɶ ɷɮɮɟɤɬɢɜɧɨɫɬɢ ɜɚɤɰɢɧɵ PC_Bt t
Ɉɬɫɸɞɚɩɨɮɨɪɦɭɥɟɭɫɥɨɜɧɨɣɜɟɪɨɹɬɧɨɫɬɢɩɨɥɭɱɚɟɦ
ij
_
PC Bt PC Bt PBt t f t
ij
D E
tt
D E
ɋɥɟɞɨɜɚɬɟɥɶɧɨ
B
PC PC Bt dt t t dt
³³
Bi j
Bij
D E
D E
Bij
D E
ij
D E
ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɩɪɟɞɵɞɭɳɟɣ ɮɨɪɦɭɥɨɣ ɢ ɪɚɜɟɧɫɬɜɨɦ
ab
ab
ɫɬɢp
n
ɧɚɣɞɢɬɟ ɩɪɨɫɬɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɭɫɥɨɜɧɨɣ ɜɟɪɨɹɬɧɨ
ab
ɉɪɟɞɩɨɥɨɠɢɦɱɬɨɩɪɟɞɜɚɪɢɬɟɥɶɧɵɟɷɤɫɩɟɪɢɦɟɧɬɵɫɜɚɤɰɢɧɨɣɞɚɸɬɨɫɧɨ
ɜɚɧɢɹɫɱɢɬɚɬɶ ɱɬɨ ɫɥɭɱɚɣɧɚɹɜɟɥɢɱɢɧɚɏ ɜɵɪɚɠɚɸɳɚɹ ɜɟɪɨɹɬɧɨɫɬɶɟɟ ɷɮɮɟɤ
ɬɢɜɧɨɫɬɢɢɦɟɟɬ ɛɟɬɚɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɩɚɪɚɦɟɬɪɚɦɢa b ȼɞɚɥɶɧɟɣɲɟɦ
ɜɚɤɰɢɧɚ ɩɪɢɦɟɧɹɥɚɫɶ ɜ ɥɟɱɟɧɢɢɞɟɫɹɬɢ ɩɚɰɢɟɧɬɨɜ ɢ ɩɨɤɚɡɚɥɚ ɫɜɨɸ ɷɮɮɟɤɬɢɜ
ɧɨɫɬɶɜɱɟɬɵɪɟɯɢɡɫɥɭɱɚɟɜ
ɍɬɨɱɧɢɬɟɩɥɨɬɧɨɫɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫ ɜ ɏ ɢ ɩɨɫɬɪɨɣɬɟ ɝɪɚɮɢɤɢɩɟɪ
ɜɨɧɚɱɚɥɶɧɨɣ ɩɥɨɬɧɨɫɬɢ ɢ ɭɬɨɱɧɟɧɧɨɣ ɇɚɣɞɢɬɟ ɜɟɪɨɹɬɧɨɫɬɶ ɷɮɮɟɤ
ɬɢɜɧɨɫɬɢɜɚɤɰɢɧɵɩɪɢɦɟɧɹɟɦɨɣɤɦɭɩɚɰɢɟɧɬɭ
ɉɪɢɦɟɪɩɨɫɬɪɨɟɧɢɹɞɜɭɯɝɪɚɮɢɤɨɜɜɨɞɧɢɯɩɪɹɦɨɭɝɨɥɶɧɵɯɤɨɨɪɞɢɧɚ
ɬɚɯ
restart: with(plots):
graph1:=plot(x^2,x=-1..1.5,color=blue):
graph2:=plot(ln(x)+1,x=-1..1.5,color=red):
display([graph1,graph2]);
ǷȏȘ
ǮȇȋȇȔȏȌ 6 ǸȚȠȌȘșȉȚȥșȈȕȒȌȌȘȒȕȍȔȢȌȉȇȗȏȇȔșȢȖȗȌȋȢȋȚȠȌȐȎȇȋȇȞȏȉȕȎȔȏȑȇȥȠȏȌ
ȉȘțȌȗȌȑȒȏȔȏȞȌȘȑȏȜȏȘȖȢșȇȔȏȐdzȕȋȏțȏȝȏȗȚȌȓȕȋȔȚȏȎȔȏȜȖȗȏȋȇȉȌȐȎȇȔȏȓȇșȌȒȣ
ȔȚȥȏȊȗȕȉȚȥțȕȗȓȚ
ɉɭɫɬɶ ɜ ɤɚɡɢɧɨ ɢɝɪɨɤ ©ɩɪɨɬɢɜɨɫɬɨɢɬª ɨɞɧɨɜɪɟɦɟɧɧɨ ɞɜɭɦ ɢɝɪɨɜɵɦ ɚɜɬɨ
ɦɚɬɚɦ Ⱥ ɢ % ©ɞɜɭɪɭɤɨɦɭ ɛɚɧɞɢɬɭª ɤɨɬɨɪɵɟ ɜ ɫɥɭɱɚɟ ɜɵɢɝɪɵɲɚ ɢɝɪɨɤɚ ɜɵ
ɩɥɚɱɢɜɚɸɬɟɦɭɡɚɨɞɢɧɪɚɭɧɞɢɝɪɵɫɨɨɬɜɟɬɫɬɜɟɧɧɨɫɜɟɪɨɹɬɧɨɫɬɹɦɢyɢzɚɜ
ɫɥɭɱɚɟɟɝɨɩɪɨɢɝɪɵɲɚ±ɧɟɜɵɩɥɚɱɢɜɚɸɬɧɢɱɟɝɨɫɨɨɬɜɟɬɫɬɜɟɧɧɨɫɜɟɪɨɹɬɧɨɫɬɹ
ɦɢ±yɢ±zȼɟɪɨɹɬɧɨɫɬɢyɢzɧɟɢɡɜɟɫɬɧɵɢɝɪɨɤɭɂɟɦɭɪɚɡɪɟɲɚɟɬɫɹɩɪɨɜɟɫɬɢ
ɫɟɪɢɸɢɡmɪɚɭɧɞɨɜɜɵɛɢɪɚɹɜɤɚɠɞɨɦɪɚɭɧɞɟɨɞɢɧɢɡɚɜɬɨɦɚɬɨɜɉɪɢɷɬɨɦɢɝ
ɪɨɤɦɨɠɟɬ ɩɪɢɦɟɧɹɬɶ ɨɞɧɭ ɢɡɫɥɟɞɭɸɳɢɯ ɞɜɭɯ ɫɬɪɚɬɟɝɢɣ ɢɝɪɵ©ɜɵɛɨɪ ɩɨ ɪɟ-
ɡɭɥɶɬɚɬɭªɢ©ɜɵɛɨɪ ɩɨ ɜɟɪɨɹɬɧɨɫɬɢª
ɋɨɝɥɚɫɧɨɫɬɪɚɬɟɝɢɢ©ɜɵɛɨɪ ɩɨ ɪɟɡɭɥɶɬɚɬɭª ɜ ɫɥɭɱɚɟ ɜɵɢɝɪɵɲɚɜkɦɪɚ
ɭɧɞɟɢɝɪɨɤɧɟɦɟɧɹɟɬɚɜɬɨɦɚɬɜɩɨɫɥɟɞɭɸɳɟɦkɦɪɚɭɧɞɟɜɫɥɭɱɚɟɩɪɨɢɝɪɵ
ɲɚɢɝɪɨɤ©ɩɟɪɟɤɥɸɱɚɟɬɫɹªɧɚɞɪɭɝɨɣɚɜɬɨɦɚɬ
Ɉɛɴɹɫɧɢɦ ɬɟɩɟɪɶ ɫɬɪɚɬɟɝɢɸ ©ɜɵɛɨɪ ɩɨ ɜɟɪɨɹɬɧɨɫɬɢª ɉɭɫɬɶ ɤ k
ɪɚɭɧɞɭɤɨɥɢɱɟɫɬɜɚ ɜɵɢɝɪɚɧɧɵɯ ɢɩɪɨɢɝɪɚɧɧɵɯɪɚɭɧɞɨɜɫɭɱɚɫɬɢɟɦ ɚɜɬɨɦɚɬɚ Ⱥ
ɪɚɜɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ wA ɢ lA ɤɨɥɢɱɟɫɬɜɚ ɜɵɢɝɪɚɧɧɵɯ ɢ ɩɪɨɢɝɪɚɧɧɵɯ ɪɚ
ɭɧɞɨɜ ɫ ɭɱɚɫɬɢɟɦ ɚɜɬɨɦɚɬɚ ȼ ɪɚɜɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ wB ɢ lB ȿɫɥɢ ɫɱɢɬɚɬɶ
ɫɥɭɱɚɣɧɵɟ ɜɟɥɢɱɢɧɵ xɢy ɪɚɫɩɪɟɞɟɥɟɧɧɵɦɢ ɪɚɜɧɨɦɟɪɧɨ ɧɚ ɨɬɪɟɡɤɟ > @
ɬ ɟ ɢɦɟɸɳɢɦɢ ɛɟɬɚɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫ ɩɚɪɚɦɟɬɪɚɦɢ D E ɬɨɜɫɢɥɭ
ɪɟɡɭɥɶɬɚɬɨɜɢɡɩɪɟɞɵɞɭɳɟɝɨɡɚɞɚɧɢɹɜɟɪɨɹɬɧɨɫɬɶɜɵɢɝɪɵɲɚɢɝɪɨɤɚɜkɦɪɚ
ɭɧɞɟɜ ɫɥɭɱɚɟ ɜɵɛɨɪɚ ɦɚɲɢɧɵȺɪɚɜɧɚ
wB
ɦɚɲɢɧɵ% ɪɚɜɧɚ
p
B
ɫɬɨɢɬ ɜ ɬɨɦ ɱɬɨ ɩɪɢ p
wB lB
! pBɢɝɪɨɤɜɵɛɢɪɚɟɬɚɜɬɨɦɚɬȺɚɩɪɢpA pB±ɚɜɬɨ
A
ɋɬɪɚɬɟɝɢɹ ©ɜɵɛɨɪ ɩɨ ɜɟɪɨɹɬɧɨɫɬɢª ɫɨ
p
A
wA lA
ɚɜ ɫɥɭɱɚɟ ɜɵɛɨɪɚ
wA
ɦɚɬȼ
Ɉɬɤɪɨɣɬɟɮɚɣɥ
ɩɚɤɟɬɚ0DSOHɜɵɧɚɣɞɟɬɟɩɪɨɰɟɞɭɪɭ
ȾɜɭɪɭɤɢɣȻɚɧɞɢɬPZVɧɚɪɚɛɨɱɟɦɥɢɫɬɟɤɨɦɩɶɸɬɟɪɧɨɝɨ
TwoArmBandit1:
restart: with(stats[random]):
TwoArmBandit1:=proc(n,m)
local win1,lose1,win2,lose2,p1,p2,new,V,y,z,strategy,ii,
average1,average2, currentmachine,i,j,k,U,a,b:
average1:=0: average2:=0:
for ii from 1 to n do
V:=uniform[0,1]: y:=V(): z:=V():
for strategy in
[strategy_on_prob,strategy_on_result] do
U:=uniform[0,1]:
win1:=0: lose1:=0: win2:=0: lose2:=0:
currentmachine:=1:
if strategy=strategy_on_prob
then
for i from 1 to m do
if currentmachine=1
then if U()<y
then win1:=win1+1:
else lose1:=lose1+1:
fi:
else if U()<z
then win2:=win2+1:
else lose2:=lose2+1:
fi:
fi:
p1:=(win1+1)/(win1+lose1+2);
p2:=(win2+1)/(win2+lose2+2);
if p1>p2
then currentmachine:=1:
else currentmachine:=2:
fi:
od:
new:=(win1+win2)/m:
average1:=average1+new:
else
for j from 1 to m do
if currentmachine=1
then if U()<y
then win1:=win1+1:
else lose1:=lose1+1:
currentmachine:=2:
fi:
else if U()<z
then win2:=win2+1:
else lose2:=lose2+1:
currentmachine:=1:
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