Probabilités : Méthodes ECE 1

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A∩B A B A B
A B
A B P (A∩B) = P(A)P(B)
P(A∩B)
(A∩B)
(A∩B)
A∪B A B A B
A B
A B A ∩B=∅P(A∪B) = P(A) + P(B)
A b P (A∪B) (A∩B)
(A∪B)
P(A∪B) = P(A) + P(B)−P(A∩B)
A B P (A∪B)
A∩B
A A A A
ΩA
P(A)=1−P(A)
A A
Ω
A P (A)6= 0 P(A)6= 1 (A, A)
n
(X=k)k X(Ω)

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k
P(Mn+1)P(Mn)
n
(P1, F1) (R1, V1, N1)

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N N
k
R R
f
R+∞
−∞ f(t)dt
RC1
k
E(aX +b) = aE(X) + b E(X+Y) = E(X) + E(Y)
X Y E(XY ) = E(X)E(Y)
E[f(X)]
X

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V(aX +b) = a2V(X)
V(X) = E(X2)−E(X)2
V(X) = EX−E(X)2
V(X)≥0E(X2)≥E(X)2
FX(x) = P(X≤x)
P(X≤k)−P(X≤k−1)
(X > x) = (X≤x)P(X > x)=1−FX(x)
P(X > x)=1−F(x)
(min(X, Y )> t)=(X > t)∩(Y > t) (max(X, Y )≤t)=(X≤t)∩(Y≤t)
X(Ω)
P(X≤k)
P(X > k)
X(Ω) = {0; 1}X
X(Ω) = J0; nKX
X(Ω) = N∗X
p
S
p
(X=k)k
X(Ω)
k

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X(Ω)
x FX(x) = P(X≤x) = 0 FX=P(X≤x)=1
x X(Ω) (X≤x) (X > x)
X
C1R
X
B(n, p)n
p
X(Ω) = J0; nKP(X=k) = n
kpk(1 −p)n−k
np np(1 −p)
n= 1)
pB(n+n0, p)
n
N >> n
n
X:= X+ 1; X
n
P(X= 0) n
Y n
P(X= 0) = P(Y= 0) = qn
P(X=k)k∈J1; nKk−1
n
n
P
k=0
xk=1−xn+1
n+1