смод

X. ɇɚ ɩɪɚɤɬɢɤɟ ɭɞɨɛɧɟɟ ɩɨɥɶɡɨɜɚɬɶɫɹ ɜɟɥɢɱɢɧɨɣ, ɪɚɡɦɟɪɧɨɫɬɶ ɤɨɬɨɪɨɣ ɫɨɜɩɚɞɚɟɬ ɫ ɪɚɡɦɟɪɧɨɫɬɶɸ ɋȼ. Ⱦɥɹ ɷɬɨɝɨ ɢɡ ɞɢɫɩɟɪɫɢɢ ɢɡɜɥɟɤɚɸɬ ɤɜɚɞɪɚɬɧɵɣ ɤɨɪɟɧɶ. ɉɨɥɭɱɟɧɧɭɸ ɜɟɥɢɱɢɧɭ ɧɚɡɵɜɚɸɬ ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɟɫɤɢɦ ɨɬɤɥɨɧɟɧɢɟɦ. ȿɟ ɨɛɨɡɧɚɱɚɸɬ ɱɟɪɟɡ VX. ɉɪɢ ɢɡɜɥɟɱɟɧɢɢ ɤɨɪɧɹ ɢɡ ɜɬɨɪɨɝɨ ɧɚɱɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɩɨɥɭɱɚɟɬɫɹ ɜɟɥɢɱɢɧɚ, ɧɚɡɜɚɧɧɚɹ ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɟɫɤɢɦ ɡɧɚɱɟɧɢɟɦ. ɑɚɫɬɨ ɢɫɩɨɥɶɡɭɸɬ

ɮɨɪɦɭɥɭ, ɫɜɹɡɵɜɚɸɳɭɸ ɨɫɧɨɜɧɵɟ ɦɨɦɟɧɬɵ:

DX = m2-mX2.

Ɍɪɟɬɢɣ ɰɟɧɬɪɚɥɶɧɵɣ ɦɨɦɟɧɬ ɫɥɭɠɢɬ ɞɥɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɚɫɢɦɦɟɬɪɢɢ (ɢɥɢ "ɫɤɨɲɟɧɧɨɫɬɢ") ɪɚɫɩɪɟɞɟɥɟɧɢɹ. ȿɫɥɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɢɦɦɟɬɪɢɱɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɆɈ, ɬɨ ɜɫɟ ɦɨɦɟɧɬɵ ɧɟɱɟɬɧɨɝɨ ɩɨɪɹɞɤɚ ɪɚɜɧɵ ɧɭɥɸ. ɉɨɷɬɨɦɭ ɟɫɬɟɫɬɜɟɧɧɨ ɜ ɤɚɱɟɫɬɜɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɚɫɢɦɦɟɬɪɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɜɵɛɪɚɬɶ ɤɚɤɨɣ-ɥɢɛɨ ɢɡ ɧɟɱɟɬɧɵɯ ɦɨɦɟɧɬɨɜ, ɢɡ ɧɢɯ ɩɪɨɫɬɟɣɲɢɣ P3. ɇɨ ɱɬɨɛɵ ɢɦɟɬɶ ɛɟɡɪɚɡɦɟɪɧɭɸ ɜɟɥɢɱɢɧɭ, ɷɬɨɬ ɦɨɦɟɧɬ ɞɟɥɹɬ ɧɚ ɤɭɛ ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɟɫɤɨɝɨ ɨɬɤɥɨɧɟɧɢɹ VX3. ɉɨɥɭɱɟɧɧɚɹ ɜɟɥɢɱɢɧɚ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɚɫɢɦɦɟɬɪɢɢ ɢɥɢ ɩɪɨɫɬɨ ɚɫɢɦɦɟɬɪɢɢ,

ɨɛɨɡɧɚɱɚɸɬ ɟɟ ɱɟɪɟɡ Sk:

Sk =.P3/VX3.

ɑɟɬɜɟɪɬɵɣ ɰɟɧɬɪɚɥɶɧɵɣ ɦɨɦɟɧɬ ɫɥɭɠɢɬ ɞɥɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɬɚɤ ɧɚɡɵɜɚɟɦɨɣ "ɤɪɭɬɨɫɬɢ" (ɨɫɬɪɨɜɟɪɲɢɧɧɨɫɬɢ ɢɥɢ ɩɥɨɫɤɨɜɟɪɲɢɧɧɨɫɬɢ) ɪɚɫɩɪɟɞɟɥɟɧɢɹ. ɗɬɢ ɫɜɨɣɫɬɜɚ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɨɩɢɫɵɜɚɸɬɫɹ ɫ ɩɨɦɨɳɶɸ ɬɚɤ ɧɚɡɵɜɚɟɦɨɝɨ ɷɤɫɰɟɫɫɚ: Ex = P4/VX4-3. ɑɢɫɥɨ 3 ɜɵɱɢɬɚɟɬɫɹ ɢɡ ɨɬɧɨɲɟɧɢɹ P4/VX4 ɩɨɬɨɦɭ, ɱɬɨ ɞɥɹ ɜɟɫɶɦɚ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɨɝɨ ɜ ɩɪɢɪɨɞɟ ɧɨɪɦɚɥɶɧɨɝɨ ɡɚɤɨɧɚ ɷɬɨ ɨɬɧɨɲɟɧɢɟ ɪɚɜɧɨ ɬɪɟɦ.

Ʉɪɨɦɟ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɢɫɩɨɥɶɡɭɸɬ ɢɧɨɝɞɚ ɚɛɫɨɥɸɬɧɵɟ ɦɨɦɟɧɬɵ (ɧɚɱɚɥɶɧɵɟ ɢ ɰɟɧɬɪɚɥɶɧɵɟ): Es=M[|X|s]; Xs=M[|X-mX|s]. ɂɡ ɧɢɯ ɱɚɳɟ ɜɫɟɝɨ ɩɪɢɦɟɧɹɸɬ ɩɟɪɜɵɣ ɚɛɫɨɥɸɬɧɵɣ ɰɟɧɬɪɚɥɶɧɵɣ ɦɨɦɟɧɬ X1=M[|X-mX|], ɧɚɡɵɜɚɟɦɵɣ ɫɪɟɞɧɢɦ ɚɪɢɮɦɟɬɢɱɟɫɤɢɦ ɨɬɤɥɨɧɟɧɢɟɦ. ȿɝɨ ɢɧɨɝɞɚ ɢɫɩɨɥɶɡɭɸɬ ɧɚɪɹɞɭ ɫɨ ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɟɫɤɢɦ ɨɬɤɥɨɧɟɧɢɟɦ ɞɥɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɪɚɫɫɟɢɜɚɧɢɹ ɋȼ, ɞɥɹ ɤɨɬɨɪɵɯ ɧɟ ɫɭɳɟɫɬɜɭɟɬ ɞɢɫɩɟɪɫɢɢ.

Ʉɪɨɦɟ ɬɚɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɢɫɩɨɥɶɡɭɸɬ ɩɨɧɹɬɢɹ ɦɨɞɚ ɢ ɦɟɞɢɚɧɚ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. Ɇɨɞɨɣ (Ɇ) ɧɚɡɵɜɚɸɬ ɧɚɢɛɨɥɟɟ ɜɟɪɨɹɬɧɨɟ ɡɧɚɱɟɧɢɟ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɦɚɤɫɢɦɭɦɭ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ (ɟɫɥɢ ɬɚɤɢɯ ɦɚɤɫɢɦɭɦɨɜ ɧɟɫɤɨɥɶɤɨ, ɬɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɧɚɡɵɜɚɸɬ ɩɨɥɢɦɨɞɚɥɶɧɵɦ). Ɇɟɞɢɚɧɚ (Ɇɟ) - ɷɬɨ ɬɚɤɨɟ ɡɧɚɱɟɧɢɟ ɋȼ X, ɞɥɹ ɤɨɬɨɪɨɝɨ P(X < Me) = P(X > Me). ȼ ɫɥɭɱɚɟ ɫɢɦɦɟɬɪɢɱɧɨɝɨ ɦɨɞɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɦɟɞɢɚɧɚ ɫɨɜɩɚɞɚɟɬ ɫ ɆɈ ɢ ɦɨɞɨɣ.

3.5.Ɉɫɧɨɜɧɵɟ ɡɚɤɨɧɵ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ, ɢɫɩɨɥɶɡɭɟɦɵɟ ɜ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɪɚɞɢɨɬɟɯɧɢɤɟ

1. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ Ʌɚɩɥɚɫɚ (ɞɜɭɯɫɬɨɪɨɧɧɢɣ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɵɣ): f(x)=O/2 exp [-O|x-P|],

ɝɞɟ P - ɆɈ; O - ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɬɟɩɟɧɶ ɪɚɡɛɪɨɫɚɧɧɨɫɬɢ X ɨɬɧɨɫɢɬɟɥɶɧɨ P.

2. Ȼɢɧɨɦɢɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ (Ȼɟɪɧɭɥɥɢ):

P(k)=CNkpk(1-p)N-k, k = 0, 1, 2,...,N.

ɇɚɩɪɢɦɟɪ, ɷɬɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɩɪɚɜɢɥɶɧɨɝɨ ɨɛɧɚɪɭɠɟɧɢɹ ɩɨ ɩɚɱɤɟ ɢɦɩɭɥɶɫɨɜ ɩɪɢ ɡɚɞɚɧɧɵɯ ɜɟɪɨɹɬɧɨɫɬɹɯ ɨɛɧɚɪɭɠɟɧɢɹ ɢ ɜɟɪɨɹɬɧɨɫɬɢ ɥɨɠɧɨɣ ɬɪɟɜɨɝɢ ɨɞɧɨɝɨ ɢɦɩɭɥɶɫɚ ɜ ɩɚɱɤɟ.

3. Ɂɚɤɨɧ ɪɚɜɧɨɦɟɪɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ.

ɉɊɂɆȿɊ. ɉɨɝɪɟɲɧɨɫɬɶ ɢɡɦɟɪɟɧɢɹ ɧɚɩɪɹɠɟɧɢɹ ɫ ɩɨɦɨɳɶɸ ɜɨɥɶɬɦɟɬɪɚ ɫ ɞɢɫɤɪɟɬɧɨɣ ɲɤɚɥɨɣ (r(a-b)/2 - ɩɨɥɨɜɢɧɚ ɞɟɥɟɧɢɹ). ɆɈ ɟɫɬɶ (a+b)/2; ɞɢɫɩɟɪɫɢɹ -

ɧɨ, Sk = 0; Ex = 0. Ⱦɥɹ ɧɨɪɦɚɥɶɧɨɝɨ ɡɚɤɨɧɚ ɩɪɢ ɧɚɯɨɠɞɟɧɢɢ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨɩɚɞɚ-

ɧɢɹ ɫɥɭɱɚɣɧɨɣ ɬɨɱɤɢ ɧɚ ɡɚɞɚɧɧɵɣ ɭɱɚɫɬɨɤ ɨɫɢ x ɢɦɟɸɬɫɹ ɬɚɛɥɢɰɵ ɬɚɤ ɧɚɡɵɜɚɟɦɨɝɨ ɢɧɬɟɝɪɚɥɚ ɜɟɪɨɹɬɧɨɫɬɟɣ; ɢɯ ɧɟɫɤɨɥɶɤɨ ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɜɵɪɚɠɟɧɢɣ, ɧɚɩɪɢ-

f

ɦɟɪ Ɏ(x)=(2S)-1/2 ³e x / 2dx (ɞɥɹ m=0 ɢ V=1). ɉɪɢ ɨɩɪɟɞɟɥɟɧɢɢ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨ-

f

ɩɚɞɚɧɢɹ ɧɚ ɭɱɚɫɬɨɤ ɨɬ a ɞɨ b ɩɨɥɭɱɢɦ P=Ɏ[(b--m)/V] - Ɏ[(a-m)/V]. ɂɧɬɟɪɟɫ ɞɥɹ ɩɪɚɤɬɢɤɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɨɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨɩɚɞɚɧɢɹ ɜ ɢɧɬɟɪɜɚɥ, ɡɚɞɚɧɧɵɣ ɜ ɟɞɢɧɢɰɚɯ ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɟɫɤɨɝɨ ɨɬɤɥɨɧɟɧɢɹ, ɧɚɩɪɢɦɟɪ, r3V. Ɍɚɤ, ɧɚɩɪɢɦɟɪ, ɷɬɚ ɜɟɪɨɹɬɧɨɫɬɶ ɟɫɬɶ 0,997. Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ ɬɚɤ ɧɚɡɵɜɚɟɦɨɟ "ɩɪɚɜɢɥɨ 3V". Ⱦɥɹ ɧɨɪɦɚɥɶɧɵɯ ɋȼ ɷɬɨ ɩɪɚɜɢɥɨ ɩɨɡɜɨɥɹɟɬ ɧɚ ɩɪɚɤɬɢɤɟ ɩɪɢɛɥɢɠɟɧɧɨ ɜɵɱɢɫɥɹɬɶ V. ɇɚɩɪɢɦɟɪ, ɩɪɢ ɨɩɪɟɞɟɥɟɧɢɢ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɞɢɚɩɚɡɨɧɚ ɦɚɝɧɢɬɨɮɨɧɚ ɫ ɩɨɦɨɳɶɸ ɨɫɰɢɥɥɨɝɪɚɮɚ ɩɪɢ ɨɬɫɭɬɫɬɜɢɢ ɜɨɥɶɬɦɟɬɪɚ.

ȼɫɟ ɨɫɬɚɥɶɧɵɟ ɡɚɤɨɧɵ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɧɟɩɪɟɪɵɜɧɵɯ ɋȼ ɨɛɪɚɡɨɜɚɧɵ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ ɪɚɜɧɨɦɟɪɧɨɝɨ ɢɥɢ ɧɨɪɦɚɥɶɧɨɝɨ ɡɚɤɨɧɨɜ, ɧɚɩɪɢɦɟɪ:

-ɡɚɤɨɧ ɋɢɦɩɫɨɧɚ (ɬɪɟɭɝɨɥɶɧɵɣ). Ⱦɢɫɩɟɪɫɢɹ D = a 2/3. ɋɜɟɪɬɤɚ ɞɜɭɯ ɪɚɜɧɨɦɟɪɧɵɯ ɡɚɤɨɧɨɜ cɨɨɬɜɟɬɫɬɜɭɟɬ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɫɭɦɦɵ ɞɜɭɯ ɧɟɡɚɜɢɫɢɦɵɯ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɵɯ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ;

-ɡɚɤɨɧ Ɋɟɥɟɹ (ɤɨɪɟɧɶ ɤɜɚɞɪɚɬɧɵɣ ɢɡ ɫɭɦɦɵ ɤɜɚɞɪɚɬɨɜ ɞɜɭɯ ɋȼ, ɪɚɫɩɪɟɞɟ-

ɥɟɧɧɵɯ ɩɨ ɧɨɪɦɚɥɶɧɨɦɭɡɚɤɨɧɭ)

f(x) = x/V2exp[-x2/(2V2)], x t 0.

Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɦɨɞɭɥɹ ɤɨɦɩɥɟɤɫɧɨɣ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ ɩɪɢ ɧɨɪɦɚɥɶɧɵɯ ɪɚɫɩɪɟɞɟɥɟɧɢɹɯ ɞɟɣɫɬɜɢɬɟɥɶɧɨɣ ɢ ɦɧɢɦɨɣ ɫɨɫɬɚɜɥɹɸɳɢɯ ɩɨɞɱɢɧɹɟɬɫɹ ɷɬɨɦɭ ɡɚɤɨɧɭ (ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɨɝɢɛɚɸɳɟɣ ɭɡɤɨɩɨɥɨɫɧɨɝɨ ɫɥɭɱɚɣɧɨɝɨ ɩɪɨɰɟɫɫɚ).

3.6.Ɉɩɪɟɞɟɥɟɧɢɟ ɡɚɤɨɧɨɜ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɋȼ ɧɚ ɨɫɧɨɜɟ ɨɩɵɬɧɵɯ ɞɚɧɧɵɯ

Ƚɢɫɬɨɝɪɚɦɦɚ

ɉɨ ɨɫɢ ɚɛɫɰɢɫɫ ɨɬɤɥɚɞɵɜɚɸɬɫɹ ɪɚɡɪɹɞɵ (ɢɧɬɟɪɜɚɥɵ ɲɢɪɢɧɨɣ l), ɢ ɧɚ ɤɚɠɞɨɦ ɢɡ ɧɢɯ ɤɚɤ ɧɚ ɨɫɧɨɜɚɧɢɢ ɫɬɪɨɢɬɫɹ ɩɪɹɦɨɭɝɨɥɶɧɢɤ, ɩɥɨɳɚɞɶ ɤɨɬɨɪɨɝɨ ɪɚɜɧɚ ɱɚɫɬɨɬɟ ɞɚɧɧɨɝɨ ɪɚɡɪɹɞɚ. Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ ɝɢɫɬɨɝɪɚɦɦɵ ɧɭɠɧɨ ɱɚɫɬɨɬɭ ɤɚɠɞɨɝɨ ɪɚɡɪɹɞɚ ɪɚɡɞɟɥɢɬɶ ɧɚ ɟɝɨ ɞɥɢɧɭ ɢ ɩɨɥɭɱɟɧɧɨɟ ɱɢɫɥɨ ɜɡɹɬɶ ɜ ɤɚɱɟɫɬɜɟ ɜɵɫɨɬɵ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɩɥɨɳɚɞɶ ɜɫɟɯ ɩɪɹɦɨɭɝɨɥɶɧɢɤɨɜ ɪɚɜɧɚ 1.

ɉɪɢ ɭɜɟɥɢɱɟɧɢɢ ɱɢɫɥɚ ɢɡɦɟɪɟɧɢɣ N ɲɢɪɢɧɭ ɢɧɬɟɪɜɚɥɨɜ l ɦɨɠɧɨ ɭɦɟɧɶɲɚɬɶ (ɭɜɟɥɢɱɢɜɚɬɶ ɢɯ ɱɢɫɥɨ m). ɉɨ ɦɟɪɟ ɭɜɟɥɢɱɟɧɢɹ N ɢ ɭɦɟɧɶɲɟɧɢɹ l ɝɢɫɬɨɝɪɚɦɦɚ ɛɭɞɟɬ ɩɪɢɛɥɢɠɚɬɶɫɹ ɤ ɝɪɚɮɢɤɭ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɜɟɥɢɱɢɧɵ X. Ɍɨ ɟɫɬɶ ɝɢɫɬɨɝɪɚɦɦɚ ɹɜɥɹɟɬɫɹ "ɩɨɪɬɪɟɬɨɦ" ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ. Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ "ɯɨɪɨɲɟɝɨ ɩɨɪɬɪɟɬɚ" ɧɟɨɛɯɨɞɢɦɨ ɩɪɢ ɡɚɞɚɧɧɨɦ N ɪɚɰɢɨɧɚɥɶɧɨ ɜɵɛɪɚɬɶ ɱɢɫɥɨ ɢɧɬɟɪɜɚɥɨɜ. ɉɪɢ ɦɚɥɨɦ ɱɢɫɥɟ ɢɧɬɟɪɜɚɥɨɜ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɛɭɞɟɬ ɨɩɢɫɵɜɚɬɶɫɹ ɫɥɢɲɤɨɦ ɝɪɭɛɨ, ɩɨ ɦɟɪɟ ɭɜɟɥɢɱɟɧɢɹ ɱɢɫɥɚ ɢɧɬɟɪɜɚɥɨɜ ɛɭɞɟɬ ɜɵɹɜɥɹɬɶɫɹ ɬɨɧɤɚɹ ɫɬɪɭɤɬɭɪɚ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ. ɇɨ ɩɪɢ ɫɥɢɲɤɨɦ ɛɨɥɶɲɨɦ ɱɢɫɥɟ ɢɧɬɟɪɜɚɥɨɜ "ɩɨɪɬɪɟɬ" ɫɧɨɜɚ ɫɭɳɟɫɬɜɟɧɧɨ ɢɫɤɚɡɢɬɫɹ - ɩɨɹɜɹɬɫɹ ɧɟɪɚɜɧɨɦɟɪɧɨɫɬɢ, ɧɟɡɚɤɨɧɨɦɟɪɧɵɟ ɞɥɹ ɢɫɫɥɟɞɭɟɦɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ (ɜ ɢɧɬɟɪɜɚɥɵ ɩɨɩɚɞɚɟɬ ɦɚɥɨ ɪɟɡɭɥɶɬɚɬɨɜ ɢɡɦɟɪɟɧɢɣ, ɢ ɷɥɟɦɟɧɬ ɫɥɭɱɚɣɧɨɫɬɢ ɩɪɢɜɨɞɢɬ ɤ ɢɫɤɚɠɟɧɢɹɦ).

ɑɢɫɥɨɜɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ

ɋɪɟɞɧɟɟ ɚɪɢɮɦɟɬɢɱɟɫɤɨɟ ɧɚɛɥɸɞɟɧɧɵɯ ɡɧɚɱɟɧɢɣ:

N

M* [X] = ¦xi/N.

i 1

ɉɪɢ ɭɜɟɥɢɱɟɧɢɢ N ɫɬɚɬɢɫɬɢɱɟɫɤɨɟ ɫɪɟɞɧɟɟ ɫɬɪɟɦɢɬɫɹ ɤ ɆɈ. Ⱥɧɚɥɨɝɢɱɧɨ ɨɰɟɧɢɜɚɟɬɫɹ ɞɢɫɩɟɪɫɢɹ - ɷɬɨ ɫɪɟɞɧɟɟ ɚɪɢɮɦɟɬɢɱɟɫɤɨɟ ɤɜɚɞɪɚɬɚ ɰɟɧɬɪɢɪɨɜɚɧɧɨɣ ɋȼ, ɬ.ɟ.

N

D* [X] = ¦(xi-mx*)2/(N-1), ɝɞɟ mx* = M* [X].

i 1

Ɍɚɤɢɦ ɠɟ ɨɛɪɚɡɨɦ ɨɩɪɟɞɟɥɹɸɬɫɹ ɞɪɭɝɢɟ ɫɬɚɬɢɫɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ, ɧɚɩɪɢɦɟɪ: ɨɩɪɟɞɟɥɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨ ɝɢɫɬɨɝɪɚɦɦɟ.

Ɂɚɞɚɱɚ ɷɬɚ ɜ ɡɧɚɱɢɬɟɥɶɧɨɣ ɦɟɪɟ ɧɟɨɩɪɟɞɟɥɟɧɧɚɹ, ɬɚɤ ɤɚɤ ɫɥɨɠɧɨ ɩɨɞɨɛɪɚɬɶ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ, ɨɬɜɟɱɚɸɳɭɸ ɦɨɞɟɥɢ ɋȼ, ɬ.ɟ., ɢɫɯɨɞɹ ɢɡ ɤɚɤɨɝɨ ɤɪɢɬɟɪɢɹ ɦɨɠɧɨ ɝɢɫɬɨɝɪɚɦɦɭ ɡɚɦɟɧɢɬɶ ɩɨɞɯɨɞɹɳɟɣ ɩɥɨɬɧɨɫɬɶɸ ɜɟɪɨɹɬɧɨɫɬɢ. Ȼɨɥɟɟ ɫɬɪɨɝɨ, ɧɨ ɫɨ ɡɧɚɱɢɬɟɥɶɧɵɦɢ ɞɨɩɭɳɟɧɢɹɦɢ ɪɟɲɚɟɬɫɹ ɷɬɚ ɩɪɨɛɥɟɦɚ ɫ ɩɨɦɨɳɶɸ ɤɪɢɬɟɪɢɟɜ ɫɨɝɥɚɫɢɹ, ɚ ɫɟɣɱɚɫ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɛɨɥɟɟ ɩɪɨɫɬɵɦɢ ɫɨɨɛɪɚɠɟɧɢɹɦɢ: ɫɧɚɱɚɥɚ ɩɪɨɢɡɜɨɞɢɦ ɚɧɚɥɢɡ ɜɢɞɚ ɝɢɫɬɨɝɪɚɦɦɵ, ɫɪɚɜɧɢɜɚɹ ɟɟ ɫ ɢɡɜɟɫɬɧɵɦɢ ɡɚɤɨɧɚɦɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ, ɚ ɡɚɬɟɦ, ɩɨɞɛɢɪɚɹ ɩɚɪɚɦɟɬɪɵ ɷɬɨɝɨ ɡɚɤɨɧɚ, ɛɭɞɟɦ ɞɨɛɢɜɚɬɶɫɹ ɧɚɢɛɨɥɶɲɟɝɨ ɫɯɨɞɫɬɜɚ ɜɢɞɚ ɫɝɥɚɠɟɧɧɨɣ ɝɢɫɬɨɝɪɚɦɦɵ ɢ ɤɪɢɜɨɣ ɢɫɩɨɥɶɡɭɟɦɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ. ɇɚɩɪɢɦɟɪ, ɟɫɥɢ ɝɪɚɮɢɤ ɫɝɥɚɠɟɧɧɨɣ ɝɢɫɬɨɝɪɚɦɦɵ ɩɨ ɜɢɞɭ ɛɥɢɡɨɤ ɤ ɧɨɪɦɚɥɶɧɨɦɭ ɡɚɤɨɧɭ, ɬɨ ɪɚɫɫɱɢɬɚɧɧɵɟ ɩɨ ɪɟɡɭɥɶɬɚɬɚɦ ɢɡɦɟɪɟɧɢɣ

ɨɰɟɧɤɢ ɆɈ ɢ Dx ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɧɨɪɦɚɥɶɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ.

4. ɋɂɋɌȿɆɕ ɋɅɍɑȺɃɇɕɏ ȼȿɅɂɑɂɇ

ɇɚ ɩɪɚɤɬɢɤɟ ɱɚɫɬɨ ɩɪɢɯɨɞɢɬɫɹ ɢɦɟɬɶ ɞɟɥɨ ɨɞɧɨɜɪɟɦɟɧɧɨ ɫ ɧɟɫɤɨɥɶɤɢɦɢ ɋȼ, ɧɚɩɪɢɦɟɪ: ɩɨɥɨɠɟɧɢɟ ɬɨɱɤɢ ɩɚɞɟɧɢɹ ɨɛɥɨɦɤɚ ɡɞɚɧɢɹ ɩɪɢ ɟɝɨ ɪɚɡɪɭɲɟɧɢɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɞɜɭɦɹ ɤɨɨɪɞɢɧɚɬɚɦɢ ɢ ɬ.ɞ. ɋɜɨɣɫɬɜɚ ɫɢɫɬɟɦɵ ɧɟɫɤɨɥɶɤɢɯ ɋȼ ɧɟ ɢɫɱɟɪɩɵɜɚɟɬɫɹ ɫɜɨɣɫɬɜɚɦɢ ɨɬɞɟɥɶɧɨɣ ɋȼ, ɬɚɤ ɤɚɤ ɩɪɢ ɷɬɨɦ ɧɟɨɛɯɨɞɢɦɨ ɨɩɢɫɚɧɢɟ ɫɜɹɡɢ ɦɟɠɞɭ ɫɨɫɬɚɜɥɹɸɳɢɦɢ ɫɢɫɬɟɦɵ ɋȼ.

4.1. Ɏɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɢɫɬɟɦɵ ɢɡ ɞɜɭɯ ɋȼ

Ɏɭɧɤɰɢɟɣ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɢɫɬɟɦɵ ɢɡ ɞɜɭɯ ɋȼ (X,Y) ɧɚɡɵɜɚɟɬɫɹ ɜɟɪɨɹɬɧɨɫɬɶ ɫɨɜɦɟɫɬɧɨɝɨ ɜɵɩɨɥɧɟɧɢɹ ɞɜɭɯ ɧɟɪɚɜɟɧɫɬɜ X < x ɢ Y < y:

F(x,y) = P((X < x)(Y < y)).

ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ, ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ F(x,y) ɟɫɬɶ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɫɥɭɱɚɣɧɨɣ ɬɨɱɤɢ (x,y) ɜ ɤɜɚɞɪɚɬ ɫ ɛɟɫɤɨɧɟɱɧɵɦɢ ɪɚɡɦɟɪɚɦɢ, ɪɚɫɩɨɥɨɠɟɧɧɵɣ ɥɟɜɟɟ ɢ ɧɢɠɟ ɷɬɨɣ ɬɨɱɤɢ ɧɚ ɩɥɨɫɤɨɫɬɢ x0y. Ɉɬɞɟɥɶɧɨ ɞɥɹ ɤɚɠɞɨɣ ɋȼ X ɢ Y ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɨɞɧɨɦɟɪɧɭɸ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ, ɧɚɩɪɢɦɟɪ, F1(x) ɟɫɬɶ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɜ ɩɨɥɭɩɥɨɫɤɨɫɬɶ, ɪɚɫɩɨɥɨɠɟɧɧɭɸ ɥɟɜɟɟ ɬɨɱɤɢ x. Ɍɚɤɠɟ ɢ F1(y) ɟɫɬɶ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɜ ɩɨɥɭɩɥɨɫɤɨɫɬɶ ɧɢɠɟ ɬɨɱɤɢ y.

ɋɜɨɣɫɬɜɚ F(x,y): 1)F(x,y) ɟɫɬɶ ɧɟɭɛɵɜɚɸɳɚɹ ɮɭɧɤɰɢɹ ɨɛɨɢɯ ɫɜɨɢɯ ɚɪɝɭɦɟɧɬɨɜ; 2)ɧɚ -f ɩɨ ɨɛɟɢɦ ɨɫɹɦ ɨɧɚ ɪɚɜɧɚ ɧɭɥɸ; 3)ɩɪɢ ɪɚɜɟɧɫɬɜɟ +f ɨɞɧɨɝɨ ɢɡ ɚɪɝɭɦɟɧɬɨɜ ɫɨɝɥɚɫɧɨ ɞɪɭɝɨɦɭ ɚɪɝɭɦɟɧɬɭ ɨɧɚ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɨɞɧɨɦɟɪɧɭɸ ɮɭɧɤɰɢɸ ɪɚɫɩɪɟɞɟɥɟɧɢɹ; 4)ɟɫɥɢ ɨɛɚ ɚɪɝɭɦɟɧɬɚ ɪɚɜɧɵ +f, ɬɨ F(x,y) = 1.

ȼɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɫɥɭɱɚɣɧɨɣ ɬɨɱɤɢ ɜ ɤɜɚɞɪɚɬ R ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ D, E ɩɨ ɨɫɢ x ɢ J, G ɩɨ ɨɫɢ y ɪɚɜɧɚ

P((X,Y) R)=F(E,G)-F(D,G)-F(E,J)+F(D,J).

F(x,y) ɫɭɳɟɫɬɜɭɟɬ ɤɚɤ ɞɥɹ ɧɟɩɪɟɪɵɜɧɵɯ, ɬɚɤ ɢ ɞɥɹ ɞɢɫɤɪɟɬɧɵɯ ɋȼ.

4.2. Ⱦɜɭɦɟɪɧɚɹ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ

Ⱦɜɭɦɟɪɧɚɹ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɟɫɬɶ ɩɪɟɞɟɥ ɫɥɟɞɭɸɳɟɝɨ ɨɬɧɨɲɟɧɢɹ:

lim {P[(X,Y) R']/('x'y)}.

'xo0,'yo0

ȿɫɥɢ F(x,y) ɧɟ ɬɨɥɶɤɨ ɧɟɩɪɟɪɵɜɧɚ, ɧɨ ɢ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚ, ɬɨ ɞɜɭɦɟɪɧɚɹ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ f(x,y) ɟɫɬɶ ɜɬɨɪɚɹ ɫɦɟɲɚɧɧɚɹ ɱɚɫɬɧɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤɰɢɢ F(x,y) ɩɨ x ɢ ɩɨ y. Ɋɚɡɦɟɪɧɨɫɬɶ f(x,y) ɨɛɪɚɬɧɚ ɩɪɨɢɡɜɟɞɟɧɢɸ ɪɚɡɦɟɪɧɨɫɬɟɣ ɋȼ X ɢ Y.

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɞɜɭɦɟɪɧɚɹ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɟɫɬɶ ɩɪɟɞɟɥ ɨɬɧɨɲɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨɩɚɞɚɧɢɹ ɜ ɦɚɥɵɣ ɩɪɹɦɨɭɝɨɥɶɧɢɤ ɤ ɩɥɨɳɚɞɢ ɷɬɨɝɨ ɩɪɹɦɨɭɝɨɥɶɧɢ-

ɤɚ, ɤɨɝɞɚ ɨɛɚ ɪɚɡɦɟɪɚ ɩɪɹɦɨɭɝɨɥɶɧɢɤɚ ɫɬɪɟɦɹɬɫɹ ɤ ɧɭɥɸ. Ƚɟɨɦɟɬɪɢɱɟɫɤɢ f(x, y) ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɧɟɤɨɬɨɪɭɸ ɩɨɜɟɪɯɧɨɫɬɶ. ȿɫɥɢ ɪɚɫɫɟɱɶ ɷɬɭ ɩɨɜɟɪɯɧɨɫɬɶ ɩɥɨɫɤɨɫɬɶɸ, ɩɚɪɚɥɥɟɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ x0y, ɢ ɫɩɪɨɟɰɢɪɨɜɚɬɶ ɩɨɥɭɱɟɧɧɨɟ ɫɟɱɟɧɢɟ ɧɚ ɩɥɨɫɤɨɫɬɶ x0y, ɬɨ ɩɨɥɭɱɢɬɫɹ ɤɪɢɜɚɹ, ɧɚɡɵɜɚɟɦɚɹ "ɤɪɢɜɨɣ ɪɚɜɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ". ɂɧɨɝɞɚ ɭɞɨɛɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫɟɦɟɣɫɬɜɚ ɤɪɢɜɵɯ ɪɚɜɧɨɣ ɩɥɨɬɧɨɫɬɢ ɩɪɢ ɪɚɡɧɵɯ ɭɪɨɜɧɹɯ ɫɟɱɟɧɢɹ.

Ʉɚɤ ɢ ɞɥɹ ɨɞɧɨɦɟɪɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɡɞɟɫɶ ɜɜɨɞɢɬɫɹ ɩɨɧɹɬɢɟ ɷɥɟɦɟɧɬɚ ɜɟɪɨɹɬɧɨɫɬɢ f(x, y)dxdy.

ȼɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɫɥɭɱɚɣɧɨɣ ɬɨɱɤɢ ɜ ɩɪɨɢɡɜɨɥɶɧɭɸ ɨɛɥɚɫɬɶ G ɨɩɪɟɞɟɥɹɟɬɫɹ ɞɜɭɦɟɪɧɵɦ ɢɧɬɟɝɪɚɥɨɦ ɨɬ f(x, y) ɩɨ ɷɬɨɣ ɨɛɥɚɫɬɢ. Ƚɟɨɦɟɬɪɢɱɟɫɤɢ ɷɬɨ ɨɛɴɟɦ, ɨɝɪɚɧɢɱɟɧɧɵɣ f(x, y) ɢ ɨɛɥɚɫɬɶɸ G. ȿɫɥɢ G ɟɫɬɶ ɩɪɹɦɨɭɝɨɥɶɧɢɤ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ ɜɟɪɲɢɧ ɩɨ ɨɫɢ x D ɢ E, ɚ ɩɨ ɨɫɢ y J ɢ G, ɬɨ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɜ ɷɬɨɬ ɩɪɹɦɨɭɝɨɥɶɧɢɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɧɬɟɝɪɚɥɨɦ

EG

P((X,Y) R)= ³³ f(x,y)dxdy.

DJ

ɋɜɨɣɫɬɜɚ ɞɜɭɦɟɪɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ:

-f(x, y) ɟɫɬɶ ɧɟɨɬɪɢɰɚɬɟɥɶɧɚɹ ɜɟɥɢɱɢɧɚ;

-ɫɜɨɣɫɬɜɨ ɧɨɪɦɢɪɨɜɤɢ ɚɧɚɥɨɝɢɱɧɨ ɨɞɧɨɦɟɪɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ, ɧɨ ɩɪɢ ɞɜɭɦɟɪɧɨɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɢ ɜ ɛɟɫɤɨɧɟɱɧɵɯ ɩɪɟɞɟɥɚɯ.

4.3. ɍɫɥɨɜɧɵɟ ɡɚɤɨɧɵ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɨɬɞɟɥɶɧɵɯ ɋȼ, ɜɯɨɞɹɳɢɯ ɜ ɫɢɫɬɟɦɭ ɋȼ

ɂɦɟɹ ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɢɫɬɟɦɵ ɞɜɭɯ ɋȼ, ɜɫɟɝɞɚ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɡɚɤɨɧɵ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɨɬɞɟɥɶɧɵɯ ɋȼ, ɜɯɨɞɹɳɢɯ ɜ ɫɢɫɬɟɦɭ. ɇɚɩɪɢɦɟɪ, F(x)=F(x,f)

ɢF(y) = F(f,y). ȿɫɥɢ ɢɡɜɟɫɬɧɚ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ f(x,y), ɬɨ F(x) = F(x,f) =

xf

³³ f(x,y)dxdy.

f f

Ⱥɧɚɥɨɝɢɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ F(y).

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɧɚɹ ɞɜɭɦɟɪɧɭɸ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ, ɜɫɟɝɞɚ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɨɞɧɨɦɟɪɧɭɸ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ. Ɉɛɪɚɬɧɭɸ ɡɚɞɚɱɭ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɪɟɲɢɬɶ ɧɟɜɨɡɦɨɠɧɨ. ȿɟ ɦɨɠɧɨ ɪɟɲɢɬɶ, ɟɫɥɢ ɢɡɜɟɫɬɧɵ ɭɫɥɨɜɧɵɟ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɢɥɢ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ.

ɍɫɥɨɜɧɵɦ ɡɚɤɨɧɨɦ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɋȼ, ɜɯɨɞɹɳɟɣ ɜ ɫɢɫɬɟɦɭ ɋȼ, ɧɚɡɵɜɚɟɬɫɹ ɟɟ ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ, ɨɩɪɟɞɟɥɟɧɧɵɣ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɞɪɭɝɚɹ ɋȼ ɩɪɢɧɹɥɚ ɨɩɪɟɞɟɥɟɧɧɨɟ ɡɧɚɱɟɧɢɟ: f(x/y). ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɧɨ ɧɚɣɬɢ ɞɜɭɦɟɪɧɭɸ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨ ɮɨɪɦɭɥɟ f(x, y) = f(x)f(y/x). ɗɬɚ ɮɨɪɦɭɥɚ ɧɚɡɵɜɚɟɬɫɹ ɬɟɨɪɟɦɨɣ ɭɦɧɨɠɟɧɢɹ ɡɚɤɨɧɨɜ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. Ɇɨɠɧɨ ɟɟ ɡɚɩɢɫɚɬɶ ɢ ɜ ɞɪɭɝɨɦ ɜɢɞɟ: f(x,y) = =f(y)f(x/y). ɂɡ ɷɬɢɯ ɜɵɪɚɠɟɧɢɣ ɫɥɟɞɭɟɬ:

f(y/x)=f(x/y)/f(x), f(x/y)=f(y/x)/f(y).

ɟɬɫɹ ɧɚ ɞɜɚ ɧɟɡɚɜɢɫɢɦɵɯ ɢɧɬɟɝɪɚɥɚ, ɜ ɤɨɬɨɪɵɯ ɢɦɟɟɬɫɹ ɩɪɨɢɡɜɟɞɟɧɢɟ ɞɜɭɯ ɩɟɪɜɵɯ ɰɟɧɬɪɚɥɶɧɵɯ ɦɨɦɟɧɬɨɜ. ɗɬɢ ɦɨɦɟɧɬɵ ɪɚɜɧɵ ɧɭɥɸ.

ɑɬɨɛɵ ɢɫɤɥɸɱɢɬɶ ɜɥɢɹɧɢɟ ɪɚɡɛɪɨɫɚɧɧɨɫɬɢ ɋȼ ɧɚ ɤɨɪɪɟɥɹɰɢɨɧɧɵɣ ɦɨɦɟɧɬ, ɟɝɨ ɞɟɥɹɬ ɧɚ ɩɪɨɢɡɜɟɞɟɧɢɟ ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɟɫɤɢɯ ɨɬɤɥɨɧɟɧɢɣ ɋȼ X ɢ Cȼ Y. ɉɨɥɭɱɚɟɬɫɹ ɛɟɡɪɚɡɦɟɪɧɚɹ ɜɟɥɢɱɢɧɚ, ɢɦɟɸɳɚɹ ɧɚɡɜɚɧɢɟ "ɤɨɷɮɮɢɰɢɟɧɬ ɤɨɪɪɟɥɹɰɢɢ":

rXY=KXY/(VXVY). ȿɫɥɢ ɋȼ X ɢ ɋȼ Y ɧɟɡɚɜɢɫɢɦɵ, ɬɨ ɜɫɟɝɞɚ rXY = 0. Ɂɧɚɱɢɬ, ɧɟɡɚɜɢɫɢɦɵɟ ɋȼ ɜɫɟɝɞɚ ɧɟɤɨɪɪɟɥɢɪɨɜɚɧɵ, ɨɞɧɚɤɨ ɨɛɪɚɬɧɨɟ ɧɟ ɜɫɟɝɞɚ ɜɟɪɧɨ. Ʉɨɪɪɟ-

ɥɢɪɨɜɚɧɧɨɫɬɶ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɧɟ ɜɫɹɤɭɸ ɜɡɚɢɦɨɡɚɜɢɫɢɦɨɫɬɶ, ɚ ɥɢɲɶ ɥɢɧɟɣɧɭɸ ɫɬɚɬɢɫɬɢɱɟɫɤɭɸ ɜɡɚɢɦɨɡɚɜɢɫɢɦɨɫɬɶ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɩɪɢ ɜɨɡɪɚɫɬɚɧɢɢ ɨɞɧɨɣ ɋȼ ɆɈ ɞɪɭɝɨɣ ɋȼ ɢɦɟɟɬ ɬɟɧɞɟɧɰɢɸ ɜɨɡɪɚɫɬɚɬɶ (ɢɥɢ ɭɛɵɜɚɬɶ) ɜ ɫɪɟɞɧɟɦ ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ. Ʉɨɷɮɮɢɰɢɟɧɬ ɤɨɪɪɟɥɹɰɢɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɬɟɩɟɧɶ ɪɚɡɛɪɨɫɚɧɧɨɫɬɢ ɤɨɨɪɞɢɧɚɬ ɬɨɱɤɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɥɢɧɟɣɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɦɟɠɞɭ X ɢ Y. ȿɫɥɢ ɋȼ X ɢ Y ɢɦɟɸɬ ɥɢɧɟɣɧɭɸ ɮɭɧɤɰɢɨɧɚɥɶɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ, ɬɨ ɤɨɷɮɮɢɰɢɟɧɬ ɤɨɪɪɟɥɹɰɢɢ ɪɚɜɟɧ r1, ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɡɧɚɤɚ ɧɚɤɥɨɧɚ ɷɬɨɣ ɮɭɧɤɰɢɢ. ɉɪɢ ɷɬɨɦ ɝɨɜɨɪɹɬ ɨ ɩɨɥɨɠɢɬɟɥɶɧɨɣ ɢɥɢ ɨɬɪɢɰɚɬɟɥɶɧɨɣ ɤɨɪɪɟɥɹɰɢɢ.

4.6. ɉɪɨɢɡɜɨɥɶɧɨɟ ɱɢɫɥɨ ɋȼ

ɑɚɫɬɨ ɩɪɢɯɨɞɢɬɫɹ ɢɦɟɬɶ ɞɟɥɨ ɜ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɪɚɞɢɨɬɟɯɧɢɤɟ ɫ ɫɢɫɬɟɦɚɦɢ ɦɧɨɝɢɯ ɋȼ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɨɥɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɫɢɫɬɟɦɵ ɋȼ ɦɨɠɟɬ ɫɥɭɠɢɬɶ ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɜɫɟɣ ɫɢɫɬɟɦɵ ɋȼ. ɇɚɩɪɢɦɟɪ, ɢɦɟɟɬɫɹ ɦɧɨɝɨɤɚɧɚɥɶɧɚɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɚɧɬɟɧɧɚɹ ɫɢɫɬɟɦɚ, ɫ ɩɨɦɨɳɶɸ ɤɨɬɨɪɨɣ ɩɪɢɟɦ ɜɟɞɟɬɫɹ ɨɞɧɨɜɪɟɦɟɧɧɨ ɜ ɧɟɫɤɨɥɶɤɢɯ ɬɨɱɤɚɯ ɩɪɨɫɬɪɚɧɫɬɜɚ. ɉɪɢ ɷɬɨɦ ɢ ɨɛɪɚɛɨɬɤɚ ɫɢɝɧɚɥɨɜ ɜ ɩɪɢɟɦɧɵɯ ɩɭɧɤɬɚɯ ɩɪɨɢɡɜɨɞɢɬɫɹ ɫɨɜɦɟɫɬɧɨ. Ⱦɥɹ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɡɚɤɨɧɨɜ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɢɫɬɟɦɵ ɛɨɥɟɟ ɱɟɦ ɬɪɟɯ ɋȼ ɩɪɢɯɨɞɢɬɫɹ ɢɫɩɨɥɶɡɨɜɚɬɶ ɦɧɨɝɨɦɟɪɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ. ɋɜɹɡɶ ɦɟɠɞɭ ɮɭɧɤɰɢɟɣ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢ ɩɥɨɬɧɨɫɬɶɸ ɜɟɪɨɹɬɧɨɫɬɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ n-ɦɟɪɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ (n - ɱɢɫɥɨ ɋȼ, ɜɯɨɞɹɳɢɯ ɜ ɫɢɫɬɟɦɭ).

ȼɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɤɨɨɪɞɢɧɚɬ ɫɥɭɱɚɣɧɨɣ ɬɨɱɤɢ ɜ ɨɝɪɚɧɢɱɟɧɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ n-ɦɟɪɧɨɣ ɫɢɫɬɟɦɵ ɨɩɪɟɞɟɥɹɟɬɫɹ n-ɤɪɚɬɧɵɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ ɩɨ ɷɬɨɦɭ ɩɪɨɫɬɪɚɧɫɬɜɭɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ.

4.7.ɑɢɫɥɨɜɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɢɫɬɟɦɵ ɧɟɫɤɨɥɶɤɢɯ ɋȼ

Ɂɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɢɫɬɟɦɵ ɋȼ (ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢɥɢ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ) ɹɜɥɹɟɬɫɹ ɩɨɥɧɨɣ, ɢɫɱɟɪɩɵɜɚɸɳɟɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɫɢɫɬɟɦɵ ɧɟɫɤɨɥɶɤɢɯ ɋȼ. Ɉɞɧɚɤɨ ɧɟ ɜɫɟɝɞɚ ɜɨɡɦɨɠɧɨ ɩɪɢɦɟɧɹɬɶ ɬɚɤɨɟ ɨɩɢɫɚɧɢɟ ɋȼ. ɇɚɩɪɢɦɟɪ, ɢɡ-ɡɚ ɨɝɪɚɧɢɱɟɧɧɨɫɬɢ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɝɨ ɦɚɬɟɪɢɚɥɚ ɢɥɢ ɢɡ-ɡɚ ɬɨɝɨ, ɱɬɨ ɬɚɤɨɟ ɨɩɢɫɚɧɢɟ ɨɛɥɚɞɚɟɬ ɢɡɥɢɲɧɟɣ ɝɪɨɦɨɡɞɤɨɫɬɶɸ. Ʉɪɨɦɟ ɬɨɝɨ, ɨɱɟɧɶ ɱɚɫɬɨ ɬɢɩ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢɡɜɟɫɬɟɧ (ɧɚɩɪɢɦɟɪ, n-ɦɟɪɧɵɣ ɧɨɪɦɚɥɶɧɵɣ). ɉɨɷɬɨɦɭɩɪɢɦɟɧɹɸɬ ɨɩɢɫɚɧɢɟ ɫɢɫɬɟɦɵ ɋȼ ɫ ɩɨɦɨɳɶɸ ɨɝɪɚɧɢɱɟɧɧɨɝɨ ɱɢɫɥɚ ɱɢɫɥɨɜɵɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ. Ʉ ɬɚɤɢɦ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦ ɨɬɧɨɫɹɬɫɹ:

-N ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɨɠɢɞɚɧɢɣ (ɆɈ), ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɯ ɫɪɟɞɧɢɟ ɡɧɚɱɟɧɢɹ ɜɯɨɞɹɳɢɯ ɜ ɫɢɫɬɟɦɭɋȼ;

-N ɞɢɫɩɟɪɫɢɣ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɯ ɫɬɟɩɟɧɶ ɢɯ ɪɚɡɛɪɨɫɚɧɧɨɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɜɨɢɯ ɆɈ;

-N(N-1) ɤɨɪɪɟɥɹɰɢɨɧɧɵɯ ɦɨɦɟɧɬɨɜ, ɨɩɪɟɞɟɥɹɸɳɢɯ ɩɨɩɚɪɧɭɸ ɤɨɪɪɟɥɹɰɢɸ

o o

ɋȼ ɜ ɫɢɫɬɟɦɟ: Kij = M[ X i Y j ], izj.

ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɤɨɪɪɟɥɹɰɢɨɧɧɵɣ ɦɨɦɟɧɬ ɩɪɢ i = j ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ

ɞɢɫɩɟɪɫɢɸ, ɬ.ɟ. Di=Kii.

ɑɚɫɬɨ ɜɫɟ ɤɨɪɪɟɥɹɰɢɨɧɧɵɟ ɦɨɦɟɧɬɵ ɪɚɫɩɨɥɚɝɚɸɬ ɜ ɜɢɞɟ ɬɚɤ ɧɚɡɵɜɚɟɦɨɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɦɚɬɪɢɰɵ:

K11 K12 ... K1N

... .

KN1 KN 2 ... KNN

ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ ɤɨɪɪɟɥɹɰɢɨɧɧɨɝɨ ɦɨɦɟɧɬɚ, Kij=Kji. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɤɨɪɪɟɥɹɰɢɨɧɧɚɹ ɦɚɬɪɢɰɚ ɜɫɟɝɞɚ "ɫɢɦɦɟɬɪɢɱɟɫɤɚɹ", ɬ.ɟ. ɟɟ ɷɥɟɦɟɧɬɵ, ɫɢɦɦɟɬɪɢɱɧɵɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɢɚɝɨɧɚɥɢ, ɪɚɜɧɵ ɦɟɠɞɭ ɫɨɛɨɣ. Ɉɛɨɡɧɚɱɚɸɬ ɟɟ ɫɢɦɜɨɥɨɦ ||Kij ||. ȼɞɨɥɶ ɝɥɚɜɧɨɣ ɞɢɚɝɨɧɚɥɢ ɩɨɫɬɚɜɥɟɧɵ ɞɢɫɩɟɪɫɢɢ. ȿɫɥɢ ɜɫɟ ɋȼ, ɜɯɨɞɹɳɢɟ ɜ ɫɢɫɬɟɦɭ ɋȼ, ɧɟɤɨɪɪɟɥɢɪɨɜɚɧɧɵ, ɬɨ ɜɫɟ ɷɥɟɦɟɧɬɵ ɦɚɬɪɢɰɵ, ɤɪɨɦɟ ɞɢɚɝɨɧɚɥɶɧɵɯ, ɪɚɜɧɵ ɧɭɥɸ. ɂɧɨɝɞɚ ɩɨɥɶɡɭɸɬɫɹ ɧɨɪɦɢɪɨɜɚɧɧɨɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɦɚɬɪɢɰɟɣ, ɫɨɫɬɚɜɥɟɧɧɨɣ ɢɡ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɤɨɪɪɟɥɹɰɢɢ: rij=Kij/(ViVj). ȿɫɥɢ ɜɫɟ ɋȼ ɧɟɤɨɪɪɟɥɢɪɨɜɚɧɧɵ, ɬɨ ɨɛɪɚɡɭɟɬɫɹ ɟɞɢɧɢɱɧɚɹ ɦɚɬɪɢɰɚ, ɭ ɤɨɬɨɪɨɣ ɞɢɚɝɨɧɚɥɶɧɵɟ ɷɥɟɦɟɧɬɵ - ɟɞɢɧɢɰɵ, ɚ ɧɟɞɢɚɝɨɧɚɥɶɧɵɟ ɷɥɟɦɟɧɬɵ - ɧɭɥɢ.

4.8. Ⱦɜɭɦɟɪɧɵɣ ɧɨɪɦɚɥɶɧɵɣ ɡɚɤɨɧ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ

Ⱦɜɭɦɟɪɧɚɹ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɡɚɞɚɟɬɫɹ ɮɨɪɦɭɥɨɣ

ɜ ɤɨɬɨɪɨɣ mX ɢ mY - ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɨɠɢɞɚɧɢɹ ɋȼ X ɢ Y; VX ɢ VY - ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɟɫɤɢɟ ɨɬɤɥɨɧɟɧɢɹ ɷɬɢɯ ɠɟ ɋȼ; R - ɤɨɷɮɮɢɰɢɟɧɬ ɤɨɪɪɟɥɹɰɢɢ.

Ɂɚɦɟɬɢɦ, ɱɬɨ ɤɪɢɜɵɟ ɪɚɜɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɢɦɟɸɬ ɜɢɞ ɷɥɥɢɩɫɨɜ:

2. Ɏɭɧɤɰɢɹ M(x) ɦɨɧɨɬɨɧɧɨ ɭɛɵɜɚɟɬ. Ɍɨɝɞɚ ɩɪɨɢɡɜɨɞɧɚɹ ɜ ɩɨɫɥɟɞɧɟɦ ɜɵɪɚɠɟɧɢɢ ɨɬɪɢɰɚɬɟɥɶɧɚ, ɧɨ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɨɬɪɢɰɚɬɟɥɶɧɨɣ. ɉɪɢɦɟɧɹɹ ɡɧɚɤ ɦɨɞɭɥɹ, ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ ɨɛɴɟɞɢɧɟɧɧɭɸ ɮɨɪɦɭɥɭ: g(y) = f(

I(y))|I'(y)|.

5.1. ɉɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɥɢɧɟɣɧɨɣ ɮɭɧɤɰɢɢ ɨɬ ɧɨɪɦɚɥɶɧɨɣ ɋȼ

ɋȼ X ɩɨɞɱɢɧɹɟɬɫɹ ɧɨɪɦɚɥɶɧɨɦɭɡɚɤɨɧɭɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ:

f(x)=(Vx 2S )-1exp[-(x-mx)2/(2Vx2)], ɚ ɋȼ Y ɫɜɹɡɚɧɚ ɫ ɧɟɣ ɥɢɧɟɣɧɨɣ ɡɚɜɢɫɢɦɨɫɬɶɸ Y = aX + b, ɝɞɟ a ɢ b - ɧɟɫɥɭɱɚɣɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ. Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ f(y).

ɉɨɫɤɨɥɶɤɭ ɡɚɞɚɧɧɚɹ ɮɭɧɤɰɢɹ ɦɨɧɨɬɨɧɧɚ, ɩɪɢɦɟɧɢɦ ɪɟɡɭɥɶɬɢɪɭɸɳɭɸ ɮɨɪɦɭɥɭ ɩɪɟɞɵɞɭɳɟɝɨ ɩɨɞɪɚɡɞɟɥɚ, ɜ ɤɨɬɨɪɨɣ I(y)=(y-b)/a,I'(y)= =1/a. ɉɨɞɫɬɚɜɢɜ ɩɨɥɭɱɟɧɧɵɟ ɮɨɪɦɭɥɵ ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɧɨɪɦɚɥɶɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ, ɩɨɥɭɱɢɦ

ɚ ɷɬɨ ɟɫɬɶ ɧɨɪɦɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ, ɭ ɤɨɬɨɪɨɣ ɆɈ mY=amX+b, ɚ ɞɢɫɩɟɪɫɢɹ VY2 = (a VX)2. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɥɢɧɟɣɧɚɹ ɮɭɧɤɰɢɹ ɨɬ ɧɨɪɦɚɥɶɧɨɝɨ ɚɪɝɭɦɟɧɬɚ ɬɚɤɠɟ ɩɨɞɱɢɧɹɟɬɫɹ ɧɨɪɦɚɥɶɧɨɦɭ ɡɚɤɨɧɭ. ɉɪɢ ɷɬɨɦ ɢɡɦɟɧɟɧɢɟ ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɟɫɤɨɝɨ ɨɬɤɥɨɧɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɧɚɱɟɧɢɟɦ ɤɨɷɮɮɢɰɢɟɧɬɚ ɚ.

5.2. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɧɟɦɨɧɨɬɨɧɧɨɣ ɮɭɧɤɰɢɢ ɨɞɧɨɣ ɋȼ

ɋɨɛɵɬɢɟ Y < y ɪɚɜɧɨɡɧɚɱɧɨ ɩɨ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨɩɚɞɚɧɢɸ ɜɨ ɜɫɟ ɭɱɚɫɬɤɢ ɧɟɨɞɧɨɡɧɚɱɧɨɫɬɢ ɩɨ ɨɫɢ x. ɉɨɷɬɨɦɭ ɜɟɪɨɹɬɧɨɫɬɢ ɫɭɦɦɢɪɭɸɬɫɹ, ɚ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ

i 'i y

ɉɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɩɨɥɭɱɚɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦ G(y) ɩɨ y. ɉɊɂɆȿɊ. ȼɵɪɚɠɟɧɢɟ y = M(x) = ax2 ɟɫɬɶ ɞɜɭɡɧɚɱɧɚɹ ɮɭɧɤɰɢɹ ɧɚ ɨɫɢ x ɨɬ -f

ɞɨ f. Ɍɨ ɟɫɬɶ ɜ ɮɨɪɦɭɥɟ (5.1) i = 2, ɱɢɫɥɨ ɫɥɚɝɚɟɦɵɯ ɪɚɜɧɨ ɬɚɤɠɟ 2 ɢ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɦɨɠɧɨ ɧɚɣɬɢ ɬɚɤ: g(y)= =[f( y / a )+f(- y / a )] 2 ay .

5.3. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɮɭɧɤɰɢɢ ɧɟɫɤɨɥɶɤɢɯ ɋȼ

ȿɫɬɶ ɫɢɫɬɟɦɚ, ɧɚɩɪɢɦɟɪ, ɢɡ ɞɜɭɯ ɋȼ X ɢ Y ɫ ɩɥɨɬɧɨɫɬɶɸ ɜɟɪɨɹɬɧɨɫɬɢ f(x, y). ɇɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɋȼ Z, ɫɜɹɡɚɧɧɨɣ ɫ X ɢ Y ɮɭɧɤɰɢɟɣ Z=M(X,Y). Ɏɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɋȼ Z ɟɫɬɶ G(z)=P(M(X,Y)<z). Ɋɚɫɫɟɱɟɦ ɩɨɜɟɪɯɧɨɫɬɶ M