Probabilités, cours, première S - MathsFG

A B
•A∩B A B
•A B A ∩B=∅A B
•A∪B
•¯ A A
P E
•A B
P(A∪B) = P(A) + P(B)−P(A∩B)
•A
P(¯ A)=1−P(A)
•
•E=A∪¯ A A ∩¯ A=∅A¯ A
P(E) = P(A∪¯ A) = P(A) +
P(¯ A)P(E)=1 1=P(A) + P(¯ A)
P(¯ A) = 1 −P(A)
A B P (A∪B) = P(A) + P(B)

E
ER
X E
I X I ={x1;x2;. . . ;xn}pi
X xi(X=xi)
X I xi
p(X=xi)
X x1x2xn
P(X=xi)P(X=x1)P(X=x2)P(X=xn)
X X
P(X= 3) = 2
6=1 3P(X=−2) = 4 6
X
X
P(X=xi)2
3
1
3
•X E(X)E(x) =
x1p1+. . . +xnpn
•X V (X)V(X) = (x1−E(X))2p1+
. . . + (xn−E(X))2pn
•X σ(X)σ(X) = pV(X)

E(X) = 2
3×(−2) + 1 3×3 = −4 3+ 1 = −1 3
a b X
•E(aX +b) = aE(X) + b
•V(aX) = a2V(X)
•aX +b ax1+b ax2+b axn+b
p1p2p3pn
E(aX +b) = p1(ax1+b) + p2(ax2 + b) + ... +pn(axn+b)
E(aX +b) = p1ax1+p1b+p2ax2+p2b+... +pnaxn+pnb
E(aX +b) = a(p1x1+p2x2+... +pnxn) + b(p1+p2+... +pn)
E(aX +b) = aE(X) + b
•E(aX) = aE(X)
V(aX) = p1(ax1−E(aX))2+p2(ax2−E(aX))2+... +pn(axn−E(aX))2
V(aX) = p1(ax1−aE(X))2+p2(ax2−aE(X))2+... +pn(axn−aE(X))2
V(aX) = a2p1(x1−E(X))2+a2p2(x2−E(X))2+... +pn(axn−aE(X))2
V(aX) = a2(p1(x2−E−X))2+p2(x2−E(X))2+... +pn(xn−E(X))2
V(aX) = a2V(X)