概率论基础讲义.pdf

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1.1 . (), , , (experiment). , , Ω . Ω , ω . Ω , A, B, C , Ω , A,B,F . Ω , , Æ σ 1.1.1 F Ω , (1) A ∈ F , AC ∈ F , AC A , AC = ¯A = Ω −A; (2) An ∈ F , n ∈ N, An ∈ F . ∞ F σ (σ ), (Ω ,F ) . n=1 , F σ , F "## $, F % #& F . : F0 = {∅, Ω}, F1 = {∅, A, AC , Ω} Æ F2 = {A∀A ⊂ Ω} σ , ’ A = {∅, A, Ω} σ . $ $! σ . A Ω . A σ , σ(A), σ(A) A " σ , A $! σ . :A = {∅, A, Ω}, σ(A) = {∅, A, AC , Ω}. +(Borel)σ : R

1 2 % (−∞, a] $! σ + σ , B, B = σ((−∞, a],∀a ∈ R). 1.1.2 (Ω ,F ) -, P F , : (/) (1) P (A) 0,∀A ∈ F ; (2) P (Ω ) = 1; (3) Ai ∈ F , i = 1, 2,· · · , AiAj = ∅,∀ i = j, ∞ ∞ () P Ai = P (Ai). () i=1 i=1 P - (Ω ,F ) %(probability measure), (probability). (Ω ,F , P ) (probability space), F &. A ∈ F , A (random event), , P (A) & A . &!&’!. % (: (1) P (∅) = 0, P (AC) = 1 − P (A). (2) A1, A2,· · · , An ), n n P Ai = P (Ai). i=1 i=1 () (3) & A Æ B, B) = P (A) + P (B) − P (AB), P (A − B) = P (A) − P (AB). P (A (4) A ⊂ B, P (A) P (B).

1.1 3 P Ai = i=1 n (5) (2 (Jordan) 3*) A1, A2,· · · , An n P (Ai) − n P (AiAj) + n · · · + (−1)n+1P (A1A2 · · · An). 1i

4 1 &%(. 1.1.1 {An, n 1} 58+ (*), lim n→∞ P (An) = P lim n→∞ An {An, n 1} 58+, % . n−1 C B1 = A1, Bn = An Ai i=1 {Bn, n 1} ). ∞ ∞ ∞ Bi, 7 Ai = i=1 i=1 P lim n→∞ An = P = P Bi i=1 Ai ∞ n n i=1 i=1 = lim n→∞ P (Bi) = lim n→∞ P = lim n→∞ P i=1 Ai = lim n→∞ P (An). = AnAC n−1, n > 1. n n Ai = i=1 i=1 ∞ n i=1 = Bi i=1 P (Bi) Bi, n 1 Æ () n An = Ai i=1 {An, n 1} 58*, {AC n , n 1} 58+, P ∞ ∞ n=1 AC n = lim n→∞ P (AC n ), ∞ C AC n = An , n=1 n=1

1 − P An n→∞(1 − P (An)), = lim 1.1 ∞ ∞ n=1 P n=1 An = lim n→∞ P (An). (+,, Borel-Cantelli &. 1.1.2 {An, n 1} &, 5 2 ∞ i=1 P (Ai) < ∞, - lim sup i→∞ Ai Ai n 58*, 7.2 1.1.1 P lim sup i→∞ Ai = 0, Ai. ∞ ∞ i=n ∞ ∞ i=n Ai = P lim n→∞ Ai = ∞ ∞ ∞ ∞ n=1 i=n i=n 0 P n=1 = lim n→∞ P Ai i=n lim n→∞ P (Ai) = 0. 2 i=n (+3&-*4, &9 2. & A, B ∈ F , P (AB) = P (A)P (B), A ’ B ) 9. 0(.21: 1 A ’ B 9; 2 A ’ BC 9; 3 P (A|B) = P (A); 4 P (A|BC) = P (A). 6& A, B, C ∈ F , P (AB) = P (A)P (B), P (AC) = P (A)P (C), P (BC) = P (B)P (C)

1 6 Æ P (ABC) = P (A)P (B)P (C), B ’ C, A, B, C )9. 2:.0: A, B, C 9, A AB ’ C, A − B ’ C )9. n & A1, A2,· · · , An ∈ F , - k(2 k n) & Ai1, Ai2,· · · , Aik(-1 i1 i2 · · · ik n), P (Ai1Ai2 · · · Aik ) = P (Ai1)P (Ai2)· · · P (Aik), A1, A2,· · · , An /. 70: A1, A2,· · · , An )9, 1 m < n, F1 = σ(Ak, 1 k m),F2 = σ(Ak, m + 1 k n). B1 ∈ F1, B2 ∈ F2, B1 ’ B2 9. & {An, n 1} )9, -)9 . ∞ P (An) = n=1 1.1.3 {An, n 1} )9&, ∞ ∞ ∞, ∞ ∞ P n=1 i=n Ai = P lim n→∞ P Ai = 1. n=1 i=n ∞ i=n Ai = lim n→∞ P ∞ i=n Ai = lim n→∞ ∞ 1−P i=n AC i .

1.2 7 ’ ∞ P i=n AC i = = P (AC i ) ∞ ∞ ∞ (1 − P (Ai)) ∞ i=n i=n − −P (Ai) e i=n (9) (1 − x e ∞ −x, x 0) P (Ai) = ∞ n i=n 2 = exp .2;. P (Ai) = 0. i=n 1.2 4!" 1. &8 34 Ω , R 9-. <5: 1.2.1 (Ω ,F , P ) ,X(ω) Ω 59 , % ∀a ∈ R , {ωX(ω) a} ∈ F , X(ω) (random variable). :7 0 (1) {ωX(ω) a} X(ω) a ω , {ωX(ω) a} (Ω ,F , P ) &, . (2) ω <<5, !, {ωX(ω) a} = {X a} = {X ∈ (−∞, a]}. ( X(ω) X, <5> ? X, Y, Z . (3) X(ω) {ωX(ω) a} ∈ F , :∀a, b ∈ R ,{X > a},{X < a},{X = a},{a < X b},{a X < b},{a < X < b},{a X b} ∈ F . 1 (Ω ,F ) F = {∅, A, AC, Ω}, A ∈ F A1 ∈ F ,

8 1 ∞ k=1 A1 IA1(ω) < {IA1 1/2} ∈ F , 7 IA1(ω) F =<5. BkBl = ∅(k = l); 2 @ (Ω ,F ) , {Bk}(0 k < ∞) Ω A, Bk = Ω Bk ∈ F (0 k < ∞) , ∞ 0,∀B ∈ B,{ω, X(ω) ∈ B} ∈ F 1 ∀a ∈ R ,{Ω , X(ω) −1(B) = (ωX(ω) ∈ xkIBk(ω) , X(ω) <5. a} ∈ F . B9 [21,22]. X = X(ω), X B). X(ω) = k=1 X (Ω ,F , P ) <5, ∀ x ∈ R , F (x) = P (X x) = P (X ∈ (−∞, x]), F (x) X 4(distribution function). <5 X -, X ()* . ∀ x ∈ R , x <5 X AC F (x), D/ f (x), F (x) = f (u) du, −∞ f (x) <5 X +%4(probability density func- tion). f (x) , dF (x) dx = f (x), P (x < X x + h) h lim h→0 = f (x), P (x < X x + h) = f (x)h + o(h).

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