Outils pour la modélisation stochastique

(Ω,A,IP) Ω A
Ω
IP
(Ω,A) (IR,B)
BIR f
fIR B A
X
IR Ω IR IP0
IIR IP0(I) = IP(X−1(I)) = IP{ω;X(ω)∈I}
X Y
A B IR
P(X∈A Y ∈B) = P(X∈A)P(Y∈B).
(X, Y ) = IE(XY )−IE(X)IE(Y).

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X= (X1, . . . , Xp)XΣ
(Xi, Xj)i j p (a1, . . . , ap)
(b1, . . . , bp)
(a1X1+ +apXp, b1X1+··· +bpXp) = (a1, . . . , ap)Σ t(b1, . . . , bp).
P Y =P X Y
PΣtP
IR IR
IR
IR
IR
IR
N(µ, σ2)
IR
f(x) = 1
√2πσ exp −(x−µ)2
2σ2
µ∈IR σ > 0
Z+∞
−∞
xpf(x)dx < +∞,
p∈IN p σ2

P(θ)
IN
IP(X=k) = exp(−θ)θk
k!
θ > 0
∞
X
k=0
kpP(X=k)<∞.
θ
X1X2
a aX1
X1X2
a aX1
n(X1, ....Xn)
σ2SnSn/√nN(0, σ2)
p(X1, . . . , Xp)p
(X1, . . . , Xp)
(a1, . . . , ap)a1X1+··· +apXp
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•

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X
ΣP
tP=P−1)DΣ = tP DP Y =tP X Y
D Y
L(X0)X0L(X1|X0)X1X0L(X2|X1, X0)X2
X1X0
n(X0, X1, .., Xn)
IN
E E = 1,2, . . . , k X
IN ×Ω
XnXn−1
IP(Xn=in|Xn−1=in−1, . . . , X0=i0) = IP(Xn=in|Xn−1=in−1).
EIP((Xn=j|Xn−1=i)
i j i j
Pnk×k