Теория вероятностей и математическая статистика. Билялов Р.Ф. - 74 стр.

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¯x =
P
x
i
n
= 0.6149, s
2
= 1.1186.
(x
1
, y
1
) (x
n
, y
n
)
m
11
=
1
n
n
P
k=1
(x
k
−
¯x)(y
k
−¯y), ¯x = (x
1
+ ... + x
n
)/n, ¯y = (y
1
+ ···+ y
n
)/n. Mm
11
Dm
11
= O(
1
n
) n → ∞.
Mx
k
= a, M y
k
= b, X
k
= x
k
−a, Y
k
= y
k
−b.
x
k
= X
k
+ a, y
k
= Y
k
+ b, ¯x =
¯
X + a, ¯y =
¯
Y + b.
m
11
=
1
n
n
X
k=1
(X
k
−
¯
X)(Y
k
−
¯
Y ) =
1
n
n
X
k=1
X
k
Y
k
−
¯
X
¯
Y .
M(X
k
Y
k
) = cov(x
1
, y
1
), M X
k
= MY
k
= 0,
M(
¯
X
¯
Y ) =
1
n
2
X
k,l
M(X
k
Y
l
) =
1
n
2
X
k6=l
MX
k
MY
k
+
1
n
2
X
k=l
M(X
k
Y
l
) =
=
1
n
2
n
X
k=1
M(X
k
Y
k
) =
1
n
cov(x
1
, y
1
).
Mm
11
=
1
n
X
cov(x
1
, y
1
) −
1
n
cov(x
1
, y
1
) = (1 −
1
n
)cov(x
1
, y
1
).
m
11
= (
1
n
−
1
n
2
)
X
k=l
X
k
Y
k
−
1
n
2
X
k,l
k6=l
X
k
Y
l
.
Dm
11
= cov(m
11
, m
11
) =
= cov
n − 1
n
2
X
k=l
X
k
Y
k
−
1
n
2
X
k,l
k
6
=
l
X
k
Y
l
,
n − 1
n
2
X
s=l
X
s
Y
s
−
1
n
2
X
s,t
s6=t
X
s
Y
t
âàðèàöèîííîãî ðÿäà, îíà òåðïèò ñêà÷îê, ðàâíûé 1/22.
                     P
                        xi
                x̄ =       = 0.6149, s2 = 1.1186.
                       n
    Çàäà÷à 9.3. Ïóñòü (x1 , y1 ),...,(xn , yn )  íåçàâèñèìûå îäèíàêîâî
                                                                    1   P
                                                                        n
ðàñïðåäåëåííûå äâóìåðíûå âåëè÷èíû. Ïîëîæèì m11 =                    n       (xk −
                                                                        k=1
x̄)(yk − ȳ), ãäå x̄ = (x1 + ... + xn )/n, ȳ = (y1 + · · · + yn )/n. Íàéòè M m11
è ïîêàçàòü, ÷òî Dm11 = O( n1 ) ïðè n → ∞.
    Ðåøåíèå. Ïîëîæèì M xk = a, M yk = b, Xk = xk − a, Yk = yk − b.
Òîãäà xk = Xk + a, yk = Yk + b, x̄ = X̄ + a, ȳ = Ȳ + b. Íàõîäèì
                          n                           n
                   1X                       1X
          m11 =       (Xk − X̄)(Yk − Ȳ ) =    Xk Yk − X̄ Ȳ .
                   n                        n
                         k=1                         k=1

Òàê êàê M (Xk Yk ) = cov(x1 , y1 ), M Xk = M Yk = 0, òî

                1 X              1 X             1 X
 M (X̄ Ȳ ) =     2
                    M (Xk Yl ) = 2   M Xk M Yk + 2   M (Xk Yl ) =
                n               n               n
                   k,l                        k6=l            k=l

                           n
                         1 X             1
                       = 2   M (Xk Yk ) = cov(x1 , y1 ).
                        n                n
                               k=1
              1X                  1                     1
    M m11   =      cov(x1 , y1 ) − cov(x1 , y1 ) = (1 − )cov(x1 , y1 ).
              n                   n                    n
                       1     1 X               1 X
               m11 = ( − 2 )        Xk Yk − 2        Xk Yl .
                      n n                     n
                                        k=l           k,l
                                                      k6=l

                               Dm11 = cov(m11 , m11 ) =
                                                                              
      n − 1 X          1 X          n−1X        1 X         
                                                            
= cov  2      Xk Y k − 2   Xk Y l ,      X Y
                                           s s −        X  Y
                                                          s t
       n              n              n2         n2 s,t      
                 k=l                 k,l             s=l
                                     k6=l                            s6=t




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