Notes du cours de statistique L3 Maths et L3 Maths

n(x1, ..., xn)∈Rn
(Ω,F,P) (X1, ..., Xn)
(Ω,F,P) (x1, ..., xn)=(X1(ω), ..., Xn(ω)) ω∈Ω
(X1, ..., Xn)
(X1,X
2, ..., Xn)
i
i−1
(X1,X
2, ..., Xn)n
P(X1, ..., Xn)
PX1, ..., Xn∼iid P
X1X2Xn
θ∈Θ Θ ⊂Rk
θ(x1, ..., xn)
Xi∼iid N(m, σ2)

i∈{1...n}θ=(m, σ)∈R∗R+∗
(X1, ..., Xn)Pθ
θ T =g(X1, ..., Xn) (X1, ..., Xn)
θ
(x1, ..., xn)t=g(x1, ..., xn)
(X1, ..., Xn)n=20
µ=E[Xi]
µ≈20
E[Xi]i
σ2=var(Xi)=E[X2
i]−E[Xi]2=E[(Xi−E[Xi])2]
σ
µ
¯
X=X1+...+Xn
nσ2
S2=n
i=1 X2
i
n−¯
X2=1
n
n
i=1
(X2
i−¯
X)2
S
¯x=x1+...+xn
ns2=n
i=1 x2
i
n−¯x2¯x=x1+...+xn n
s2
¯x=20.075 s2=0.072875
2s=0.26995
π
n= 1000
xi=0
xi=1
(x1, ..., xn) (X1, ..., Xn)
θ=π=P[Xi=1]
π F 1
(X1, ..., Xn)
F=card{i∈{1...n}|Xi=1}
n=n
i=1 Xi
n
π=E[Xi]F=¯ X
(x1, ..., xn)π
f=card{i∈{1...n}|xi=1}
n=n
i=1 xi
n=0.48

Xi¯
XF S
xif¯x s
π µ σ
(X1, ..., Xn)
L(θ;x1, ..., xn)=P(X1=x1, ..., Xn=xn;θ)=n
i=1 P(Xi=xi;θ)Xi
L(θ;x1, ..., xn)=n
i=1 f(xi;θ)Xif(xi,θ)
l(θ;x1, ..., xn)=ln(L(θ;x1, ..., xn))
(x1, ..., xn)Pθt θ θ
θ→L(θ;x1, ..., xn)
t=argmaxθL(θ;x1, ..., xn)
L(θ;x1, ..., xn)
(x1, ..., xn)θ
(X1, ..., Xn)θ=π
1−πsix
i=0
Pθ(Xi=xi)=
πsix
i=1
θ=π
Pθ(Xi=xi)=πxi(1 −π)1−xipour xi∈{0,1}
(x1, ..., xn)∈{0,1}n
L(π;x1, ..., xn)=
n
i=1
Pθ(Xi=xi)
=
n
i=1
πxi(1 −π)1−xi
=πn
i=1 xi(1 −π)n−n i=1 xi

l(π;x1, ..., xn)=ln(π)
n
i=1
xi+ln(1 −π)(n−
n
i=1
xi)
∂l(π;x1, ..., xn)
∂π =n
i=1 xi
π(1 −π)−n 1−π
∂l(π;x1,...,xn)
∂π >0n
i=1 xi
n>π
n
i=1 xi
nF=n
i=1 Xi
n
(X1, ..., Xn)N(µ, σ2)
θ=(µ, σ)
(X1, ..., Xn)
µ σ Xixi∈R
fθ(xi)= 1
√2πσ exp −(xi−µ)2
2σ2
θ=(µ, σ) (x1, ..., xn)∈Rn
L(θ;x1, ..., xn)=
n
i=1
fθ(xi)
=
n
i=1
1
√2πσ exp −(xi−µ)2
2σ2
=1
(2π)n/2σnexp −n
i=1(xi−µ)2
2σ2
µ=¯x
σ=s
¯x=15.9679 s=1.7846