le sujet - Université d`Angers

θ=e−λ=Pθ(X= 0) X1, ..., Xn
P(λ)
b
θ1=e−X
b
θ2=1
n
n
X
i=1
(Xi= 0)
b
θ3=1−1
nPn
i=1 Xi
n= 10 λ= 1
b
θ1b θ2b θ3
λ∈[0.1; 6]
E b θ2
n
X
i=1
Xi!=b θ3.
X f h X
R|h(x)|f(x)dx < ∞
I=Zh(x)f(x)dx =Ef[h(X)]
X1, . . . , Xn
f
lim
n→∞
ˆ
In= lim
n→∞
1
n
n
X
i=1
h(Xi) = I, ,
1
nPn
i=1 h(Xi)I
XΓ(a, b)
fa,b(x) = baxa−1
Γ(a)exp(−bx)Ix>0
a= 4 b= 1

1000 X
X
X2 5
85 90 95% X
π
n[0,1]×[0,1]
(Sn)n∈NS0= 0
∀n≥1
Sn=Sn−1+Xn,
(Xn)n≥1P(X1= 1) = p= 1 −P(X1=−1) = 1 −q
ρ=q
p(Sn)Z
n∈Np∈]0,1[
n(Sn)
n∈N∗k≤n
0n T {0, n}
ET=k(n−k)p=1
2;ET=n(1 −ρk) (p−q)(1 −ρn)−k p−qp6=q.
P(ST= 0) = n−k
np=1 2;P(ST= 0) = ρk−ρn 1−ρnp6=q.