Vikipediya ro'yxatidagi maqola
Ushbu iboralarda,
ϕ ( x ) = 1 2 π e − 1 2 x 2 { displaystyle phi (x) = { frac {1} { sqrt {2 pi}}} e ^ {- { frac {1} {2}} x ^ {2}}} bo'ladi standart normal ehtimollik zichligi funktsiyasi,
Φ ( x ) = ∫ − ∞ x ϕ ( t ) d t = 1 2 ( 1 + erf ( x 2 ) ) { displaystyle Phi (x) = int _ {- infty} ^ {x} phi (t) , dt = { frac {1} {2}} left (1+ operator nomi {erf} chap ({ frac {x} { sqrt {2}}} o'ng) o'ng)} mos keladi kümülatif taqsimlash funktsiyasi (qayerda erf bo'ladi xato funktsiyasi ) va
T ( h , a ) = ϕ ( h ) ∫ 0 a ϕ ( h x ) 1 + x 2 d x { displaystyle T (h, a) = phi (h) int _ {0} ^ {a} { frac { phi (hx)} {1 + x ^ {2}}} , dx} bu Ouenning T funktsiyasi .
Ouen[nb 1]
Aniq bo'lmagan integrallar
∫ ϕ ( x ) d x = Φ ( x ) + C { displaystyle int phi (x) , dx = Phi (x) + C} ∫ x ϕ ( x ) d x = − ϕ ( x ) + C { displaystyle int x phi (x) , dx = - phi (x) + C} ∫ x 2 ϕ ( x ) d x = Φ ( x ) − x ϕ ( x ) + C { displaystyle int x ^ {2} phi (x) , dx = Phi (x) -x phi (x) + C} ∫ x 2 k + 1 ϕ ( x ) d x = − ϕ ( x ) ∑ j = 0 k ( 2 k ) ! ! ( 2 j ) ! ! x 2 j + C { displaystyle int x ^ {2k + 1} phi (x) , dx = - phi (x) sum _ {j = 0} ^ {k} { frac {(2k) !!} { (2j) !!}} x ^ {2j} + C} [nb 2] ∫ x 2 k + 2 ϕ ( x ) d x = − ϕ ( x ) ∑ j = 0 k ( 2 k + 1 ) ! ! ( 2 j + 1 ) ! ! x 2 j + 1 + ( 2 k + 1 ) ! ! Φ ( x ) + C { displaystyle int x ^ {2k + 2} phi (x) , dx = - phi (x) sum _ {j = 0} ^ {k} { frac {(2k + 1) !! } {(2j + 1) !!}} x ^ {2j + 1} + (2k + 1) !! , Phi (x) + C} Ushbu integrallarda, n !! bo'ladi ikki faktorial : hatto uchun n u 2 dan to barcha juft sonlarning ko'paytmasiga teng n va g'alati uchun n u 1dan to toqgacha bo'lgan barcha toq sonlarning ko'paytmasi n ; qo'shimcha ravishda shunday deb taxmin qilinadi 0!! = (−1)!! = 1 .
∫ ϕ ( x ) 2 d x = 1 2 π Φ ( x 2 ) + C { displaystyle int phi (x) ^ {2} , dx = { frac {1} {2 { sqrt { pi}}}} Phi left (x { sqrt {2}} ) o‘ngda) + C} ∫ ϕ ( x ) ϕ ( a + b x ) d x = 1 t ϕ ( a t ) Φ ( t x + a b t ) + C , t = 1 + b 2 { displaystyle int phi (x) phi (a + bx) , dx = { frac {1} {t}} phi left ({ frac {a} {t}} right) Phi chap (tx + { frac {ab} {t}} o'ng) + C, qquad t = { sqrt {1 + b ^ {2}}}} [nb 3] ∫ x ϕ ( a + b x ) d x = − 1 b 2 ( ϕ ( a + b x ) + a Φ ( a + b x ) ) + C { displaystyle int x phi (a + bx) , dx = - { frac {1} {b ^ {2}}} left ( phi (a + bx) + a Phi (a + bx) ) o'ng) + C} ∫ x 2 ϕ ( a + b x ) d x = 1 b 3 ( ( a 2 + 1 ) Φ ( a + b x ) + ( a − b x ) ϕ ( a + b x ) ) + C { displaystyle int x ^ {2} phi (a + bx) , dx = { frac {1} {b ^ {3}}} left ((a ^ {2} +1) Phi ( a + bx) + (a-bx) phi (a + bx) right) + C} ∫ ϕ ( a + b x ) n d x = 1 b n ( 2 π ) n − 1 Φ ( n ( a + b x ) ) + C { displaystyle int phi (a + bx) ^ {n} , dx = { frac {1} {b { sqrt {n (2 pi) ^ {n-1}}}}}} Phi chap ({ sqrt {n}} (a + bx) o'ng) + C} ∫ Φ ( a + b x ) d x = 1 b ( ( a + b x ) Φ ( a + b x ) + ϕ ( a + b x ) ) + C { displaystyle int Phi (a + bx) , dx = { frac {1} {b}} chap ((a + bx) Phi (a + bx) + phi (a + bx) o‘ngda) + C} ∫ x Φ ( a + b x ) d x = 1 2 b 2 ( ( b 2 x 2 − a 2 − 1 ) Φ ( a + b x ) + ( b x − a ) ϕ ( a + b x ) ) + C { displaystyle int x Phi (a + bx) , dx = { frac {1} {2b ^ {2}}} left ((b ^ {2} x ^ {2} -a ^ {2 } -1) Phi (a + bx) + (bx-a) phi (a + bx) o'ng) + C} ∫ x 2 Φ ( a + b x ) d x = 1 3 b 3 ( ( b 3 x 3 + a 3 + 3 a ) Φ ( a + b x ) + ( b 2 x 2 − a b x + a 2 + 2 ) ϕ ( a + b x ) ) + C { displaystyle int x ^ {2} Phi (a + bx) , dx = { frac {1} {3b ^ {3}}} left ((b ^ {3} x ^ {3} +) a ^ {3} + 3a) Phi (a + bx) + (b ^ {2} x ^ {2} -abx + a ^ {2} +2) phi (a + bx) right) + C } ∫ x n Φ ( x ) d x = 1 n + 1 ( ( x n + 1 − n x n − 1 ) Φ ( x ) + x n ϕ ( x ) + n ( n − 1 ) ∫ x n − 2 Φ ( x ) d x ) + C { displaystyle int x ^ {n} Phi (x) , dx = { frac {1} {n + 1}} left ( left (x ^ {n + 1} -nx ^ {n-) 1} o'ng) Phi (x) + x ^ {n} phi (x) + n (n-1) int x ^ {n-2} Phi (x) , dx right) + C } ∫ x ϕ ( x ) Φ ( a + b x ) d x = b t ϕ ( a t ) Φ ( x t + a b t ) − ϕ ( x ) Φ ( a + b x ) + C , t = 1 + b 2 { displaystyle int x phi (x) Phi (a + bx) , dx = { frac {b} {t}} phi left ({ frac {a} {t}} right) Phi chap (xt + { frac {ab} {t}} o'ng) - phi (x) Phi (a + bx) + C, qquad t = { sqrt {1 + b ^ {2} }}} ∫ Φ ( x ) 2 d x = x Φ ( x ) 2 + 2 Φ ( x ) ϕ ( x ) − 1 π Φ ( x 2 ) + C { displaystyle int Phi (x) ^ {2} , dx = x Phi (x) ^ {2} +2 Phi (x) phi (x) - { frac {1} { sqrt) { pi}}} Phi chap (x { sqrt {2}} o'ng) + C} ∫ e v x ϕ ( b x ) n d x = e v 2 2 n b 2 b n ( 2 π ) n − 1 Φ ( b 2 x n − v b n ) + C , b ≠ 0 , n > 0 { displaystyle int e ^ {cx} phi (bx) ^ {n} , dx = { frac {e ^ { frac {c ^ {2}} {2nb ^ {2}}}} {b { sqrt {n (2 pi) ^ {n-1}}}}} Phi left ({ frac {b ^ {2} xn-c} {b { sqrt {n}}}} o'ng) + C, qquad b neq 0, n> 0} Aniq integrallar
∫ − ∞ ∞ x 2 ϕ ( x ) n d x = 1 n 3 ( 2 π ) n − 1 { displaystyle int _ {- infty} ^ { infty} x ^ {2} phi (x) ^ {n} , dx = { frac {1} { sqrt {n ^ {3} ( 2 pi) ^ {n-1}}}}} ∫ − ∞ 0 ϕ ( a x ) Φ ( b x ) d x = 1 2 π | a | ( π 2 − Arktan ( b | a | ) ) { displaystyle int _ {- infty} ^ {0} phi (ax) Phi (bx) dx = { frac {1} {2 pi | a |}} chap ({ frac {) pi} {2}} - arctan chap ({ frac {b} {| a |}} o'ng) o'ng)} ∫ 0 ∞ ϕ ( a x ) Φ ( b x ) d x = 1 2 π | a | ( π 2 + Arktan ( b | a | ) ) { displaystyle int _ {0} ^ { infty} phi (ax) Phi (bx) , dx = { frac {1} {2 pi | a |}} left ({ frac {) pi} {2}} + arctan chap ({ frac {b} {| a |}} o'ng) o'ng)} ∫ 0 ∞ x ϕ ( x ) Φ ( b x ) d x = 1 2 2 π ( 1 + b 1 + b 2 ) { displaystyle int _ {0} ^ { infty} x phi (x) Phi (bx) , dx = { frac {1} {2 { sqrt {2 pi}}}}} chap (1 + { frac {b} { sqrt {1 + b ^ {2}}}} o'ng)} ∫ 0 ∞ x 2 ϕ ( x ) Φ ( b x ) d x = 1 4 + 1 2 π ( b 1 + b 2 + Arktan ( b ) ) { displaystyle int _ {0} ^ { infty} x ^ {2} phi (x) Phi (bx) , dx = { frac {1} {4}} + { frac {1} {2 pi}} chap ({ frac {b} {1 + b ^ {2}}} + arctan (b) right)} ∫ 0 ∞ x ϕ ( x ) 2 Φ ( x ) d x = 1 4 π 3 { displaystyle int _ {0} ^ { infty} x phi (x) ^ {2} Phi (x) , dx = { frac {1} {4 pi { sqrt {3}} }}} ∫ 0 ∞ Φ ( b x ) 2 ϕ ( x ) d x = 1 2 π ( Arktan ( b ) + Arktan 1 + 2 b 2 ) { displaystyle int _ {0} ^ { infty} Phi (bx) ^ {2} phi (x) , dx = { frac {1} {2 pi}} left ( arctan ( b) + arctan { sqrt {1 + 2b ^ {2}}} o'ng)} ∫ − ∞ ∞ Φ ( a + b x ) 2 ϕ ( x ) d x = Φ ( a 1 + b 2 ) − 2 T ( a 1 + b 2 , 1 1 + 2 b 2 ) { displaystyle int _ {- infty} ^ { infty} Phi (a + bx) ^ {2} phi (x) , dx = Phi left ({ frac {a} { sqrt) {1 + b ^ {2}}}} o'ng) -2T chap ({ frac {a} { sqrt {1 + b ^ {2}}}}, { frac {1} { sqrt { 1 + 2b ^ {2}}}} o'ng)} ∫ − ∞ ∞ x Φ ( a + b x ) 2 ϕ ( x ) d x = 2 b 1 + b 2 ϕ ( a t ) Φ ( a 1 + b 2 1 + 2 b 2 ) { displaystyle int _ {- infty} ^ { infty} x Phi (a + bx) ^ {2} phi (x) , dx = { frac {2b} { sqrt {1 + b ^ {2}}}} phi chap ({ frac {a} {t}} o'ng) Phi chap ({ frac {a} {{ sqrt {1 + b ^ {2}}} { sqrt {1 + 2b ^ {2}}}}} o'ng)} [nb 4] ∫ − ∞ ∞ Φ ( b x ) 2 ϕ ( x ) d x = 1 π Arktan 1 + 2 b 2 { displaystyle int _ {- infty} ^ { infty} Phi (bx) ^ {2} phi (x) , dx = { frac {1} { pi}} arctan { sqrt {1 + 2b ^ {2}}}} ∫ − ∞ ∞ x ϕ ( x ) Φ ( b x ) d x = ∫ − ∞ ∞ x ϕ ( x ) Φ ( b x ) 2 d x = b 2 π ( 1 + b 2 ) { displaystyle int _ {- infty} ^ { infty} x phi (x) Phi (bx) , dx = int _ {- infty} ^ { infty} x phi (x) Phi (bx) ^ {2} , dx = { frac {b} { sqrt {2 pi (1 + b ^ {2})}}}}} ∫ − ∞ ∞ Φ ( a + b x ) ϕ ( x ) d x = Φ ( a 1 + b 2 ) { displaystyle int _ {- infty} ^ { infty} Phi (a + bx) phi (x) , dx = Phi left ({ frac {a} { sqrt {1 + b) ^ {2}}}} o'ng)} ∫ − ∞ ∞ x Φ ( a + b x ) ϕ ( x ) d x = b t ϕ ( a t ) , t = 1 + b 2 { displaystyle int _ {- infty} ^ { infty} x Phi (a + bx) phi (x) , dx = { frac {b} {t}} phi left ({ frac {a} {t}} right), qquad t = { sqrt {1 + b ^ {2}}}} ∫ 0 ∞ x Φ ( a + b x ) ϕ ( x ) d x = b t ϕ ( a t ) Φ ( − a b t ) + 1 2 π Φ ( a ) , t = 1 + b 2 { displaystyle int _ {0} ^ { infty} x Phi (a + bx) phi (x) , dx = { frac {b} {t}} phi left ({ frac { a} {t}} o'ng) Phi chap (- { frac {ab} {t}} o'ng) + { frac {1} { sqrt {2 pi}}} Phi (a) , qquad t = { sqrt {1 + b ^ {2}}}} ∫ − ∞ ∞ ln ( x 2 ) 1 σ ϕ ( x σ ) d x = ln ( σ 2 ) − γ − ln 2 ≈ ln ( σ 2 ) − 1.27036 { displaystyle int _ {- infty} ^ { infty} ln (x ^ {2}) { frac {1} { sigma}} phi left ({ frac {x} { sigma }} o'ng) , dx = ln ( sigma ^ {2}) - gamma - ln 2 taxminan ln ( sigma ^ {2}) - 1.27036} Adabiyotlar
Patel, Jagdish K.; O'qing, Kempbell B. (1996). Oddiy tarqatish bo'yicha qo'llanma (2-nashr). CRC Press. ISBN 0-8247-9342-0 CS1 maint: ref = harv (havola ) Ouen, D. (1980). "Normal integrallar jadvali". Statistikadagi aloqa: simulyatsiya va hisoblash . B9 : 389–419. CS1 maint: ref = harv (havola ) 
