TP Modélisation Statistique

m×n
p1, p2, ...
(−3,3)
95%
P N x ∈ {1, ..., N}m
m n N P
x1, ..., xm
p n1, ..., npnt

X F
F−
F−(u) = inf{x∈R|F(x)≥u}.
U∼ U[0,1] F−(U)F
F F (X) [0,1]
n= 500 λ= 2
f g M
f≤Mg (Yn)n≥1g(Un)n≥1
[0,1] Yi
T= inf n≥1|Un≤f(Yn)
Mg(Yn)
X:= YTf
n= 500
N(0,1)
n= 200 R2
Y1Y2f1
f2
X p Y1Y2
Y=Y1X+Y2(1 −X).
φµ,σ2N(µ, σ2)n= 1000
f(x) = 1
3φ−2,1(x) + 2 3φ3,4(x), x ∈R.
(X, Y )R2
f(x, y) = 1
2πexp(−x2+y2
2), x, y ∈R.
(X, Y )X=Rcos(T)Y=Rsin(T)
R∈[0,1] T∈[0,2π)
R T
(U, V ) [0,1]
(X, Y ) = p−2 ln(U) cos(2πV ),p−2 ln(U) sin(2πV ),
R2

n= 1000
[0,2]
k k
1 1000
5
N= 1000 n= 50
2
X1, ..., Xnn= 200
N(0,1)
Fn
F
k1n
Yk=kFk−Fk∞= sup
x∈R|Fk(x)−F(x)|,
k
N= 1000
Ynn= 200
N(0,1)
I1=ZR e−2x2|cos(x)|dx
I2=ZD ex2cos(y)q1 + log2(y)dxdy, D ={(x, y)∈[0,1]2:x+ sin(y)≤1}

X1, ..., XkN(0,1)
X= (X1, ..., Xk)>RkX∼ N(0,I)
B∈Rp×km∈RpY=BX+m∼ N(m,BB>)
Y1
√2πppdet(Σ) exp −1
2(y−m)>Σ−1(y−m), y ∈Rp,
Σ = BB>B
Y
m=2
3,Σ = 1 1 1 3 ,
χ2Rn
χ2n χ2(n)
χ2(n)χ2(n0)
χ2(n+n0)
Y∼ N(m,Σ) f f(Y)∼χ2(2)
X Y N(0,1)
χ2(n)
T=X
pY/n
nT(n)
TnT(n)
nT(n)
n= 1,5,10
X Y χ2(n)χ2(n0)
F=X/n
Y/n0
n n0F(n, n0)
n, n0F(n, n0)

X= (X1, ..., Xn)>N(m, σ2)
RX=1
nPn
i=1 Xi
1= (1, ..., 1)>X−X.1 1 Rn
1
σ2Pn
i=1(Xi−X)2Rn−1
X−X.1X
T=√nX−m
qPn
i=1(Xi−X)2/(n−1)
.
n= 10,m= 1, σ2= 4
P4
P4P10
n= 10000 P
ˆ
P P
ˆ
P
P5
n= 10000
ˆ
Q
ˆ
Qij =Pn
k=2 1{Xk=i, Xk−1=j}
Pn
k=2 1{Xk=i}.
ˆ
Qij Qij := Pji π(j)
π(i)π
Q
10
A B A n Xn(n+1)