Cor2 - Jean-Romain HEU

N
(Xk)k∈N∗f
F
N∈N∗RN= maxk6NXkRNN
RNFR=FN
x∈RFR(x) = P(R6x) = P(maxXk6x) = P(X16x∩···∩XN6x)
Xk
FR(x) = P(X16x)× ··· × P(XN6x) = FN(x).
Xk
[0,1]
RN
f(x) = 1 [0,1]
F(x) = x[0,1] f
FR(x) = xN[0,1] RNfR(x) =
F0
R(x) = NxN−1
RN
E(RN) = R1
0xfR(x)dx =R1 0NxNdx =N
N+1
N
RN
RNN
N1< N2RN2>RN2
Xk(RN)

l61
l= 1 RN
RN
RN
a < 1
a < 1ε > 1P(l6a) = ε
RNl
NP(RN6a)>ε RN
P(RN> a)6E(RN)
aP(RN6a)=1−P(RN< a)>E(RN) a
E(RN)a < 1NP(RN6a)>1
X1a
a
k > 1pa=P(Xk> a)f F
pa=P(Xk> a) = 1 −F(a) = Z+∞
a
f(x)dx.
Y a
n>1Y=n∀k∈ {2,3, . . . , n}Xk6a Xn+1 > a
Y
Y
paY∼ G(pa)
a
E(Y) = 1
pa
b
pbpa
b b > a F F (b)>
F(a)pb6pa
a
1
pab
1
pb>1 pac>b
c1
pc>1 pb

f
XE(X)=0 V(X)≈10
±√10 0,5−0,5
f
XE(X)≈0V(X)≈10 P(X62) ≈0,75
−a, 0, a
0,25−0,5−0,25 a
a=√5
{−3,2,4} {0.5,0.25,0.25}
(Xk)k>1E(λ)
λ
λ Xk
f(x) = λe−λx x>0
1
λ
1
λ2
n>100 Xnn
Xk
Xn=Pn
k=1 Xk
n.
Xnn+∞
1/λ Xk
λ= 0,5n= 100
P(1,96X100 62,1).
nXn−1/λ
1
λ√n
λ n
P(1,96X100 62,1) = P(−0,56Xn−2
0,260,5) ≈2FN(0,5) −1≈0,4.
αnβnn
P(αn
λ6Xn6βn λ)≈0,99.
n√n(λXn−1) ∼ N(0,1)
Xn

FN(2,57) ≈0,995
P(−2,57 6√n(λXn−1) 62,57) ≈0,99.
P(1−2,57/√n
λ6Xn61+2,57/√n λ)≈0,99.
αn= 1 −2,57/√n
βn= 1 + 2,57/√n
λ n Xk
Xn1% λ
[αn
Xn,βn Xn]
X100 = 2,23
λ
α100
X100 =1−0,257
2,23 ≈0,33 β100
X100 =1+0,257
2,23 ≈0,56 1% λ∈[0,33; 0,56]
λ0,05
λ < 0,8
n
Xn
n λ
1% 0,05
99% Xn
Xnn > 166
αn
λ>1−2,57/√166 0,8>1n > 166 Xn
n[αn
Xn,βn Xn] 0,05
n > 166 βn−αn
Xn=2×2,57/√n Xn<5,14
√nn > 10600