STATISTIQUE - Exercices Divers

X
X
p(X=x)
p(|X| ≥ 1)
{Xn}Sn=1
n
n
P
k=1
X2
k
Xn∼ N(0,1) Sn
Xn∼ L(µ, σ2), Sn
k k ≥100
A

X
Y=X/n
Y
n
pµX
n≥0.44¶= 0.03
X1, X2, ..., X8λ= 2
P(X1+X2+... +X8≥32)
Z=X1+X2+... +X8
P(Z≥32) ≤E(Z)/32 = 1/2
λ E(X) = λ V ar(X) = λ
X1, X2, ..., X8V ar(Z) = V ar(X1) + ... +V ar(X8) = 16
P(Z≥32) = P(Z−EZ ≥16) ≤
P(|Z−EZ| ≥ 16) ≤V ar(Z)/162= 1/16
X1, X2, ..., X81
2
n= 500 p=1
2m=E[X] = np = 5001 2= 250
σ2=V ar[X] = np(1 −p) = 5001
2
1
2= 125
P(250 −10 ≤X≤250 + 10) = P(−10
σ≤Y≤10 σ) = 0.6289066305...
P(250 −10 ≤X≤250 + 10) = P(−10.5
σ≤Y≤10.5 σ) = 0.6523455198...
P(240 ≤X≤260) = 0.6523358207.
n= 1000 p= 0,04 P(X > 30)
np = 40
np(1 −p) = 38.4
P(X > 30) ≈P(Y > 30) = P(Y−40
√38.4>30−40 √38.4≈0.947
Z= (Y−40)/√38.4
np = 40
P(X > 30) ≈0.938

σ2= 25.
P(X > 85) ≤E[X]/85 = 15/17 ∼
=0.88
P(X > 85) = P(X2>852)≤E[X2]/852= (σ2+E[X2]
852) = (25+752) 852=226 289 ∼
=0.78.
P(65 ≤X≤85) = P(|X−E[X]| ≤ 10) = 1 −P(|X−E[X]|>10) ≥1−σ2
102=3 4
Mn =(X1+X2+...+X n)
n
σ2
n=25 nP(|Mn −
E[Mn]| ≤ 5) = 1 −P(|Mn −E[Mn]|>5) ≥1−25
(52n)= 1 −1 n,3 4
P(|Mn −E[M n]| ≤ 5) ∼
=P(−n1
2≤Z≤n1 2)
P(−n1
2≤Z≤n1 2)n= 1 ,0.6826894920 ; n=
2,0.8427007929 ; n= 3 ,0.9167354835 ; n4,0.9544997360 ; n= 5 ,0.9746526813
n= 3
(n≈10)
17.8
7.1
17.8+7.1 = 24.9
λ= 24.9
λ60 = 60 ∗24.9 = 1494.
m=E[T] = λ60 = 60∗24.9 = 1494 σ2:= V ar[T] =
λ60 = 1494.
Z=(T−m)
σZ∼N(0,1). P (X > 1500) =
P(X≥1500.5) = P((X−1494)
√1494 ≥(1500.5−1494) √1494 'P(Z≥(1500.5−1494) √1494 '0.4332263648
fX(x) = x
20< x < 2

X1, ..., XNfX
ZN=(X1+...+XN)
N
P(|Z−4
3|>10−3)≤0.01
E[X] = 4
3E[X2] = 2 V ar[X] = 2 9
E[ZN] = 4
3V ar[ZN] = V ar[X] N=2 9N
P(|ZN−4
3|>0.001) ≤2∗106 9NN >
22222222.
X45
10
350
X∼P(λ)Y=X−λ
√λY
Ti
T=T1+... +T20 20