ED Probabilité Statistique

p∈Nn, N, q ∈N∗
n
X
k=0
Ck
n= 2n,
n
X
k=0
kCk
n=n2n−1,
n
X
k=0
Ck
pCn−k q=Cn p+q,
n
X
k=0 Ck
n2=Cn 2n, nCq−1
n−1=qCq
n, Cp n+Cp+1 n=Cp+1
n+1,
N
X
k=n
Cn
k=Cn+1
N+1.
2n∈Nn2r
r≤n2r
2r
2r k 1≤k≤r
(i, j, k, l, m)∈N5i+j+k+l+m= 1000
30 1 365
Ω = {(a, b, c)∈N3,1≤a<b<c≤50 c−b=b−a}
P
A={(a, b, c)∈Ω, b −a= 10}
B={(a, b, c)∈Ω, b −a > 2}
C={(a, b, c)∈Ω, c 6= 50}
Ω
a c b

A B P(A∪B)P(A)P(B)P(A∩B)
(An)n∈[[1,N]]
P
[
n∈[[1,N]]
An
=
N
X
n=1
(−1)n−1X
1<i1<i2<···<in
P
\
k∈[[1,n]]
Aik
.
PTk∈[[1,n]] Aikn
P
[
n∈[[1,N]]
An
=
N
X
n=1
(−1)n−1Cn
NP n
\
k=1
Ak!.
N N
N→+∞r n r ≥N
0,5%
ρ= 0,95
ρ95%

f:E7→ FE ⊂ P(F)E
E
X X
A B B A
f g h := sup(f, g)
f|f|
|f|f
1AA
µ(N,P(N)) fN R+
Rfdµ Pn∈Nf(n)
RAf(x)dµ(x) = R(1A(x)f(x))dµ(x)f
f∀n∈N, f(n) = +∞λR+
Rfdλ = +∞
fR R+µRF(u) =
Ru
−∞ f(x)dµ(x)
(0,1)
B(]0,1[) ]0,1[
]0,1[ ]0,1[
]r−δ, r +δ[r δ ]0,1[
B(]0,1[)
C1={[a, b], a ≤b, a, b ∈]0,1[}
C1={[a, b], a ≤b, a, b ∈]0,1[∩Q}
C1={]0, t], t ∈]0,1[}
C2={]0,1/2n[,]k/2n, k/2n+1[, n ∈N, k ∈N,1≤k≤2n−1}
n∈N]0,1[
Bn:= σ0,1
2n,k 2n,k 2n+1 , k ∈N,1≤k≤2n−1.
(Bn)n∈N∪nBn
f=1[0,1]∩Qf(x)=1 x∈[0,1] ∩Qf(x)=0
f[0,1]
[0,1] ∩Q(rn)n∈N∗
fn=1r1,···rnn≥1fn[0,1] (fn)
f
f f

ZR
sin(x)µ(dx)µ(dx) = X
n∈N
δπn(dx)
ZR
x3µ(dx)µ(dx) = 1
√2πe−x2
2λ(dx), λ R
ZR
1[1,2](x)exsin(πx/2)µ(dx)µ(dx) = δ2(dx) + e−xλ(dx).
lim
n→+∞Z+∞
0
e−nsin2(x)f(x)dx f
lim
n→+∞Zn
01−x
nnex/2dx, lim
n→+∞Zn
0
ln(x)1−x
nndx.
f:x7→ R+∞
0
e−xt
1+t2dt
f x ∈R+
fR∗
+f00(x) + f(x)=1/x f
f(x, y)=1/(1 −xy)
R[0,1]2f(x, y)dx dy =Pn≥11/n2
x= (u+v)/√2y= (u−v)/√2Pn≥11/n2=π2/6

X
X
X
E(X)
Var (X)
X
k0 1 2 3 4 5
P(X=k) 0,05 0,15 0,2 0,35 0,15 0,1
X
Y
Y X
E(X)
Var (X)
a∈R+X{1,2,3,4,5}
P(X=k) = 1
6a(k−a)2.
a
n
X
k=1
k2=n(n+ 1)(2n+ 1)
6
n
X
k=1
k3=n(n+ 1)
22
E(X) Var (X)
n∈N∗1n
k
k k
X1X2
X1
X2
n
X
k=1
P(X2=k)=1.
E(X2) = 1−n
2+3n+ 1 2
n
X
k=1
1
n+k.
k/(n+k) = k−n+n2/(n+k)E(X2)
n→+∞