Loi de Bernoulli et loi binomiale, cours, première S - MathsFG

p p ∈[0; 1] X X
p
•X
•P(X= 1) = p P (X= 0) = 1 −p
X p
•E(X) = p
•V(X) = p(1 −p)
E(X)=0×P(X= 0) + 1 ×P(X= 1) = 0 ×(1 −p)+1×p=p
V(X) = P(X= 0) ×(0 −E(X))2+P(X= 1) ×(1 −E(X))2= (1 −p)p2+p(1 −p)2=
p(1 −p)(p+ 1 −p) = p(1 −p)
p
p
t
t < p

P
() BT
T < P
”P”? →P
→T
T < P
p
n n p
•

p= 0,05
p= 0,05 n= 3
•
n= 3 p= 0,05
P(SS ¯ S) = 0,052×0,95 ≈0,002
n k k ≤nn
kk n
k n −k n
n
k=n! k!(n−k)!
n! = n×(n−1) ×(n−2) ×... ×2×1

3
3= 1
3
2= 3
SSE SES ESS
p∈[0; 1] n n ≥1
X
X n
p X ∼ B(n;p)
X n p
k k ∈ {0; 1; 2; ...;n}P(X=k) = n
kpk(1 −p)n−k
n
kk n
pk(1 −p)n−k
P(X=k)X
n= 3 p= 0,05 X
n= 3 p= 0,05

P(X= 2) = P(SS ¯ S∩S¯ SS ∩¯ SSS)
P(X= 2) = P(SS ¯ S) + P(S¯ SS) + P(¯ SSS)
P(X= 2) = 0,052×0,95 + 0,95 ×0,05 ×0,95 + 0,95 ×0,052= 3 ×0,052×0,95 = 0,007
P(X=k)P(X≤k)
X n p k n P (X=k)
P(X≤k)
•
P(X=k)P(X≤k)
•
k n p
P(X=k)
P(X≤k)
P(X=k)P(X≤k)X
X n = 4 p= 0,4
P(X≤3)
•P(X < 3) = P(X≤2)
•P(X > k) 1 −P(X≤k)
X n p
•E(X) = np
•V(X) = np(1 −p)
n p