A Lea To Ire 67
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1 g(x) = [(2π)n det(K)]−1/2 exp({− (x − µ)t K −1 (x − µ)}) 2
Φη (t) = E[exp(iht, ηi)] =
d Y
Φηi (ti ) =
i=1
d Y
e
−t2i /2
=e
−||t||2 /2
.
i=1
ΦX (t) = E[exp(iht, Xi)] = E[exp i(ht, Aη + µi)] = E[exp i(ht, Aηi)]eiht,µi t
= E[exp i(hA t, ηi)]e
iht,µi
t
= Φη (A t)e
iht,µi
=e
−||At t||2 /2 iht,µi
e
X M
L
M
L
M[
||At t||2 = hAt t, At ti = hAAt t, ti = hKX t, ti.
z
KX .
E(X)
X
\
W
[M
V
[
\
M[
L
\V
[
\
[
XW
\
[M
L
M
M
XW
U
[M
V
XW
M
O
O
e
e
d
dc
b
`a
z
\V
P
[M
P
L
XM
[
X
X
X
\
W
Z
M
L
V
[
P
M
V
[
P
X[W
\
M
XW
Z
V
N
\ M
X
\
\ "
$
^
i
U1 , . . . U n
hX, Ui iUi , KX Ui = λi Ui , Var(hX, Ui i) = λi hX − E(X), Ui i) √ ηi = . λi ]
SX
M
M
P
M
\L
XM
XM
L
[
LM
M
N
\VV
X L
MV $
^
η
cov(ηi , ηj ) = p
M
N
[
X = Aη + µ
1 λi i j p hKX U , U i = δij . λi λj λi λj
V
L
M O
√ λ 1 U1 1 A= ... √ λ1 U1 n
...
√
... ...
λn Un 1
... √
λn Un n
, µ = E(X)
z
X=
X
KX
KX
e
e
d
dc
b
`a
+
WL
O
U
M Z [
WL
γ(x)dx
\
^
X = φ(η),
N M[ P
M
XM
X
T[
O ⊂ Rn K
M N
2π
L
$
M
XW
. n
√
PM W P
L
V
[M
T
\ \ XW
\V
[
P
XM
WL
[
L
M
K
[
−||x||2 /2
e γ(x) =
z
γ(φ−1 (y))|Jac(φ−1 )(y)|dy =
Z
φ−1 (O)
P(X ∈ O) = P(η ∈ φ−1 (O)) =
Z
φ : Rn →Rn , x→φ(x) = Ax + µ
e
e
d
dc b `a
d+
M
P
\V
XM
M
N
φ−1 (y) = A−1 (y − µ). )
W
||φ−1 (y)||2 = hA−1 (y − µ), A−1 (y − µ)i = hA−1t A−1 (y − µ), (y − µ)i = hK −1 (y − µ), (y − µ)i.
1 . Jac(φ−1 ) = det(A−1 ) = p det(K)
z
X : g(y) = γ(φ−1 (y))|Jac(φ−1 )(y)|.
⇒
e
e
d
dc
b
`a +
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