1 Notion d expérience aléatoire Définition 1.1 Expérience aléatoire Exemple 1.1 Remarqe. Définition 1.2 Événement Remarqe.

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1 Ω = {, 2, 3, 4, 5, 6} A A A B A B A B A B A B A B Ω A B A B A {A, A} Ω {{ω}, ω Ω}

2 A B A B A B A B C {A, B, C} A B A B A B A B = 2652 (7, V ) (V, 7 ) (7, V ) (V, 7 ) ( 52 2 ) = = = 90 (n, n) = 36 (, 6) (6, ) (, 6) (6, ) ( 6 2) + 6 = 2

3 Ω P : P(Ω) [0, ] P(Ω) = A B P(A B) = P(A) + P(B) (Ω, P) Ω P Ω (Ω, P) P( ) = 0 (A, B) P(Ω) 2 P(A B) = P(A) + P(B) P(A B) A P(Ω) P(A) = P(A) (A, B) P(Ω) 2 A B P(A) P(B) S A S P(A) = Ω P Ω P({ω}) = ω Ω (p ω ) ω Ω ω Ω p ω = P Ω P({ω}) = p ω ω Ω

4 Ω = {a, b} P Ω P({a}) = 0 P({b}) = (Ω, P) S P(Ω) A S P(A) = Ω Ω Ω P({ω}) = ω Ω Ω Ω P Ω A P(A) = A Ω A P(A) = 4 52 = 3 B P(B) = 3 52 = 4 (Ω, P) (A, B) P(Ω) 2 P(B) 0 A B P(A B) P B (A) P(A B) P(B)

5 00 B + B + P(B + ) = 7 00 A B R + P(B) = = P(R+) = 00 = B P(B R + 7 ) = = 7 75 P(B+ ) = P(B R + ) = P(R + )P(B R + ) B P(R + B) = = 7 9 P(B+ ) = P(B R + ) = P(B)P(R + B) (Ω, P) B P(Ω) P(B) 0 P B : A P(Ω) P B (A) Ω (Ω, P) (A,..., A n ) P(Ω) n P(A A n ) 0 P(A A n ) = P(A )P(A 2 A )P(A 3 A A 2 )... P(A n A A n ) B i i P(B B 2 B 3 ) P(B ) = 3 0 B P(B 2 B ) = 2 0 B B P(B 3 B B 2 ) = 0 P(B B 2 B 3 ) = P(B )P(B 2 B )P(B 3 B 2 B ) = = 3 500

6 (Ω, P) S B P(B) = A S P(B A)P(A) S A P(B) = P(B A)P(A) + P(B A)P(A) T M P(M) = P(T M) = 00 P(T M) = 00 P(T) = P(T M)P(M) + P(T M)P(M) = ( 00 7 ) = (Ω, P) A B P(A) 0 P(B) 0 P(A B) = P(B A)P(A) P(B) S A B P(B) 0 P(A B) = P(B A)P(A) P(B S)P(S) S S A B A B B A B AV AR BV BR A P(AV BR) P(AV BR) = P(BR AV) P(BR AV)P(AV) + P(BR AR)P(AR) P(AV) = 3 5 P(AR) = 2 5 P(BR AV) = 3 6 P(BR AR) = 4 6 P(AV BR) = = 5 7

7 (Ω, P) A B P(A B) = P(A)P(B) (Ω, P) A B P(B) > 0 A B P(A B) = P(A) B B 2 B B 2 B B 2 (Ω, P) (A i ) i I (A i ) i I J I, P A j = P(A j ) j J j J (A i ) i I A i

8 A A 2 A 3 P(A ) = P(A 2 ) = P(A 3 ) = 2 P(A A 2 ) = P(A A 3 ) = P(A 2 A 3 ) = 4 A A 2 A 3 P(A A 2 A 3 ) = 0 P(A )P(A 2 )P(A 3 ) A A 2 A Ω Ω X(Ω) R S X(Ω) = 2, 2 (Ω, P) X Ω A X(Ω) X (A) Ω X (A) X A A {x} X ({x}) X = x P(X A) P(X = x) X x R P(X ], x]) P(X [x, + [) P(X x) P(X x) X Ω X = x x X(Ω) P(X = x) = x X(Ω)

9 X Ω = {PPP, FPP, PFP, PPF, FFP, FPF, PFF, FFF} X = X ({}) {FPP, PFP, PPF} X 2 X ([2, + [) X ({2, 3}) {FFP, FPF, PFF, FFF} (Ω, P) X Ω X P X : { P(X(Ω)) [0, ] A P(X A) X Y X Y X L X L (Ω, P) X Ω P X X(Ω) (Ω, P) X Ω P X P(X = x) x X(Ω) X Ω P(X A) A P(X(Ω)) P(X = x) x X(Ω) (Ω, P) X Ω X E x E, P(X = x) = E (Ω, P) X Ω {0, } X p [0, ] P(X = ) = p P(X = 0) = p X B(p)

10 X X() = X() = 0 p = P() (Ω, P) A Ω A P(A) (Ω, P) X Ω 0, n X n N p [0, ] P(X = k) = ( n k) p k ( p) n k k 0, n X B(n, p) n p B(n, p) q r n X ( ) ( Y ) X B n, Y B q q+r n, r q+r (Ω, P) X Ω f : X(Ω) E Y = f X Ω Y = f(x) Y = f X (Ω, P) { X Ω f : X(Ω) E Y = f(x) P(Y(Ω)) [0, ] A P(X f (A)) y Y(Ω) P(Y = y) = P(X f ({y})

11 X 2, 2 Y = X 2 {0,, 4} P(Y = 0) = P(X = 0) = 5 P(Y = ) = P(X = ) + P(X = ) = 2 5 P(Y = 4) = P(X = 2) + P(X = 2) = 2 5 (Ω, P) X { Ω B P(Ω) P(B) > 0 P(X(Ω)) [0, ] X B P X B : A P(X A B) X B P(X = x B) x X(Ω) P(X A B) A P(X(Ω)) X Y Ω (X, Y) { Ω X(Ω) Y(Ω) ω (X(ω), Y(ω)) (Ω, P) X Y Ω X Y (X, Y) P (X,Y) (A, B) P(X(Ω)) P(Y(Ω)), P (X,Y) (A B) = P((X, Y) A B) X Y P((X, Y) = (x, y)) (x, y) X(Ω) Y(Ω) (X, Y) P((X, Y) = (x, y)) (x, y) X(Ω) Y(Ω) (X, Y) (X, Y) X Y (Ω, P) X Y Ω X Y x X(Ω), P(X = x) = P((X, Y) = (x, y)) y Y(Ω), P(Y = y) = y Y(Ω) x X(Ω) P((X, Y) = (x, y))

12 2 4 4 U V U V Ω = {{, 2}, {, 3}, {, 4}, {2, 3}, {2, 4}, {3, 4}} U {, 2, 3} V {2, 3, 4} U V (Ω, P) X Y Ω y Y(Ω) X Y = y P X Y=y x X(Ω) Y X = x P Y X=x n (Ω, P) X{,..., X n Ω Ω X (Ω) X (X,..., X n ) n (Ω) ω (X (ω),..., X n (ω)) X,..., X n (X,..., X n ) X,..., X n (Ω, P) X Y Ω X Y (A, B) P(X(Ω)) P(Y(Ω)), P((X, Y) A B) = P(X A)P(Y B) (Ω, P) X Y Ω X Y (x, y) X(Ω) Y(Ω), P((X, Y) = (x, y)) = P(X = x)p(y = y)

13 X Y (X, Y) X Y A B A B (Ω, P) (X i ) i I (X i ) i I (A i ) i I ( ) P(X i (Ω)), P X i A i = P(X i A i ) i I i I i I (Ω, P) (X i ) i I (X i ) i I (x i ) i I ( ) X i (Ω), P X i = x i = P(X i = x i ) i I i I i I X Y {, } Z = XY X, Y, Z (A i ) i I ( Ai ) i I (Ω, P) p [0, ] X,..., X n B(p) X + + X n B(n, p) n p B(n, p) (Ω, P) X Y Ω f g X(Ω) Y(Ω) f(x) g(y)

14 (Ω, P) X Ω X E(X) = P({ω})X(ω) = P(X = x)x ω Ω x X(Ω) X X X X (Ω, P) A P(Ω) E( A ) = P(A) (Ω, P) X Ω X E(X) = 0 X X E(X) (Ω, P) X Y Ω (λ, µ) R 2 E(λX + µy) = λe(x) + µe(y) X Ω X E(X) 0 X Y Ω X Y E(X) E(Y) X E(X) E( X ) R Ω (Ω, P) X Ω c R E(X) = c X E R E(X) = E x p [0, ] X Ω B(p) E(X) = p n N p [0, ] X Ω B(n, p) E(X) = np x E

15 (Ω, P) X Ω f X(Ω) R E(f(X)) = P({ω})f(X(ω)) = P(X = x)f(x) ω Ω x X(Ω) X, n E(2 X ) (X, Y) f X(Ω) Y(Ω) R E(f(X, Y)) = f(x, y)p(x = x, Y = y) (x,y) X(Ω) Y(Ω) = x X(Ω) y Y(Ω) = y Y(Ω) x X(Ω) f(x, y)p(x = x, Y = y) f(x, y)p(x = x, Y = y) (Ω, P) X Y Ω X Y E(XY) = E(X)E(Y) X {, } Y = X (Ω, P) X Ω k N k X E ( X k) k X E ( (X E(X)) k) (Ω, P) X Y Ω X Y (X, Y) = E ((X E(X)) (Y E(Y))) = E(XY) E(X)E(Y)

16 (Ω, P) X Y Ω (X, Y) = (Y, X) X Y Z Ω (λ, µ) R 2 (λx + µy, Z) = λ (X, Z) + µ (Y, Z) (X, λy + µz) = λ (X, Y) + µ (Y, Z) X Ω (X, X) 0 R Ω (Ω, P) X Y Ω X Y (X, Y) = 0 X Y { X P(X = 0) = P(X = ) = P(X = ) = 3 Y = X = 0 0 X 0 (X, Y) = 0 X Y Y X P((X, Y) = (0, 0)) = 0 P(X = 0)P(Y = 0) 0 (X, Y) = 0 X Y (Ω, P) X Ω X ( V(X) = (X, X) = E (X E(X)) 2) = E(X 2 ) E(X) 2 X σ(x) = V(x) 2 V(X) = P({ω}) (X(ω) E(X)) 2 = P(X = x) (x E(X)) 2 ω Ω ( ) V(X) = P({ω})X(ω) 2 E(X) 2 = ω Ω x X(Ω) x X(Ω) P(X = x)x 2 E(X) 2

17 (Ω, P) X Y Ω V(X + Y) = V(X) + V(Y) + 2 (X, Y) V(X Y) = V(X) + V(Y) 2 (X, Y) (X + Y, X Y) = V(X) V(Y) (X, Y) = 2 (V(X + Y) V(X) V(Y)) = 2 (V(X) + V(Y) V(X Y)) = (V(X + Y) V(X Y)) 4 (Ω, P) X Y Ω (X, Y) V(X) V(Y) = σ(x)σ(y) (X,Y) σ(x)σ(y) X Y (Ω, P) X,..., X n Ω V(X + + X n ) = V(X ) + + V(X n ) (Ω, P) X Ω c R V(X) = 0 p [0, ] X Ω B(p) V(X) = p( p) n N p [0, ] X Ω B(n, p) V(X) = np( p) (Ω, P) X Ω X V(X) = (Ω, P) X Ω (a, b) R 2 V(aX + b) = a 2 V(X) (Ω, P) X Ω σ(x) > 0 X E(X) σ(x)

18 (Ω, P) X Ω a R + X P(X a) E(X) a P( X a) E( X ) a (Ω, P) X Ω α R + P ( X E(X) ε) V(X) ε 2 = σ(x)2 ε 2 X,..., X n m σ X n = n X k V( n X n ) = σ2 n ε R + k= P ( Xn m ε ) σ2 nε 2 P ( Xn m ε ) = 0 n +