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x1, ..., xn
X
X
θ
X
ˆ
θn=ˆ
θn(X1, ..., Xn),
θˆ
θn
θ(x1, ..., xn)
θ
ˆ
θn(x1, ..., xn).
n
(x1, ..., xn)
•(X1, ..., Xn) = (x1, ..., xn). •f
f(x1)...f(xn).
Ln(θ)
n(x1, ..., xn)θ

ˆ
θn(x1, ..., xn)
(x1, ..., xn)X
λ
λ
Ln(λ) = e−nλλx1+...+xn
x1!...xn!.
λ
e−nλλx1+...+xn
ˆ λn=1
n
n
X
i=1
xi.
ˆ
θnθ
θn∈N∗
ˆ
θnθ
ˆ λn=1
nPn
i=1 xi
ˆ
θnθ
θn→ ∞
ˆ
θnθ
(x1, ..., xn)
µ
ˆµn(x1, ..., xn) = 1
n
n
X
i=1
xi.
n x1, ..., xn

(x1, ..., xn)Xα1, ..., αk
p1, ..., pk
p=(p1, ..., pk)
ˆpi=1
n
n
X
j=1 {αi}(xj), i = 1, ..., k.
pi
αi
X
µ V (x1, ..., xn)
X(ˆµ, ˆ
V)
ˆµ=1
n
n
X
i=1
xiˆ
V=1
n−1
n
X
i=1
(xi−ˆµ)2.
n
n−1
n−1
nV
n
n−1
µ
ˆ
V=1
n
n
X
i=1
(xi−µ)2
ˆσ=pˆ
V,
ˆ
VV
((x1, y1), ..., (xn, yn))
(X, Y )
d
cov cov(X, Y )
d
cov =1
n−1
n
X
i=1
(xi−ˆµ)(yi−ˆν),
ˆµˆν X
Y
d
corr corr(X, Y )
d
corr =d
cov
ˆσ.ˆτ,
ˆσˆτ X
Y

ˆ
θnθ
δ
ˆ
θθ
θ∈[ˆ
θn±δ]
α
[ˆ
θn±δ]α
(x1, ..., xn)
µN(µ, 1)
α
ˆµn(x1, ..., xn) = 1
n
n
X
i=1
xi,
µ
1
n√n(ˆµn−µ)
δ>0
P r |ˆµn−µ| ≤ δ
√n!=1
√2πZ|x|≤δe−x2/2dx.
α δ
1
√2πZ|x|≤δe−x2/2dx =α.
µˆµ−δ/√n , ˆµ+δ/√n
α
α
Xµ
1(x1, ..., xn)X
ˆµn=1
nPn
i=1 xnµ
αˆµn
ˆµn
n→ ∞ √n(ˆµn−µ)
P r |ˆµn−µ| ≤ δ
√n!≍1
√2πZ|x|≤δe−x2/2dx.
δ
1
√2πZ|x|≤δe−x2/2dx =α.
µˆµ−δ/√n , ˆµ+δ/√n
α
α