Теория вероятностей и математическая статистика. Билялов Р.Ф. - 89 стр.

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a
σ
2
a σ
2
a τ
n−1
=
(
¯
X−a)
√
n−1
√
m
2
,
n − 1
n = 22,
¯
X = 0.6149, m
2
= 1.0678. τ
0.025,21
= 2.08,
p
m
2
/(n −1) =
0.2256,
p
m
2
/(n − 1)τ
α,n−1
= 0.469. 0.146 <
a < 1.084.
σ
2
m
2
n/σ
2
, χ
2
n−1
.
χ
2
0.99,21
= 8.9, χ
2
0.05,21
= 32.7. 22m
2
/χ
2
0.05,21
= 0.7181,
22m
2
χ
2
0.99,21
= 2.6395. P (0.7181 < σ
2
< 2.6395) = 0.94.
X
1
, X
2
, ..., X
n
H
0
: P (X
k
= i) = p
0
i
> 0; H
1
: P (X
k
= i) = p
1
i
> 0; i = 1, ..., N;
n
X
i=1
p
(0)
i
=
n
X
i=1
p
(1)
i
= 1.
L
H
0
(X
1
, X
2
, ..., X
n
) = p
(0)
1
m
1
p
(0)
2
m
2
···p
(0)
N
m
N
, m
k
X = k. L
H
1
(X
1
, X
2
, ..., X
n
) = p
(1)
1
m
1
p
(2)
2
m
2
···p
(1)
N
m
N
.
ln(L
H
1
/L
H
0
) = m
1
ln(p
(1)
1
/p
(0)
1
) + m
2
ln(p
(1)
2
/p
(0)
2
) + ···m
N
ln(p
(1)
N
/p
(0)
N
).
X
1
, X
2
, ..., X
15
P (X
k
= l) = p
l
, l = 1, 2, 3.
p
l
= p
(0)
l
= 1/3, l = 1, 2, 3, p
1
= p
(1)
1
= 0, 40,
p
2
= p
(1)
2
= 0, 42, p
3
= p
(1)
3
= 0, 18.
α
3.7.3    10-îå ïðàêòè÷åñêîå çàíÿòèå. Äîâåðèòåëüíûé èíòåð-
        âàë. Ñòàòèñòè÷åñêàÿ ïðîâåðêà ãèïîòåç
Çàäà÷à 9.13. Ïî âûáîðêå, ïîëó÷åííîé â çàäà÷å 9.2, ïîñòðîèòü äîâå-
ðèòåëüíûé èíòåðâàë äëÿ a ñ äîâåðèòåëüíîé âåðîÿòíîñòüþ 0.95 è äëÿ
σ 2 ñ äîâåðèòåëüíîé âåðîÿòíîñòüþ 0.94.
    Ðåøåíèå. a è σ 2 íåèçâåñòíû. Äëÿ ïîñòðîåíèÿ äîâåðèòåëüíîãî          √ èí-
                                                                  (X̄−a) n−1
òåðâàëà äëÿ a âîñïîëüçóåìñÿ ñëó÷àéíîé âåëè÷èíîé τn−1 =                √
                                                                       m2    ,
ïîä÷èíåííîé ðàñïðåäåëåíèþ Ñòüþäåíòà ñ n − 1 ñòåïåíÿìè     p         ñâîáîäû,
n = 22,pX̄ = 0.6149, m2 = 1.0678. τ0.025,21 = 2.08, m2 /(n − 1) =
0.2256, m2 /(n − 1)τα,n−1 = 0.469. Äîâåðèòåëüíûé èíòåðâàë: 0.146 <
a < 1.084.
    Äëÿ ïîñòðîåíèÿ äîâåðèòåëüíîãî èíòåðâàëà äëÿ σ 2 èñïîëüçóåì
ñëó÷àéíóþ âåëè÷èíó m2 n/σ 2 , ïîä÷èíåííóþ ðàñïðåäåëåíèþ χ2n−1 . Èç
òàáëèöû íàõîäèì: χ20.99,21 = 8.9, χ20.05,21 = 32.7. 22m2 /χ20.05,21 = 0.7181,
22m2 χ20.99,21 = 2.6395. P (0.7181 < σ 2 < 2.6395) = 0.94.
    Çàäà÷à 9.15. Íàéòè ñòàòèñòèêó íàèáîëåå ìîùíîãî êðèòåðèÿ,
ðàçëè÷àþùåãî ïî âûáîðêå X1 , X2 , ..., Xn ãèïîòåçû

  H0 : P (Xk = i) = p0i > 0; H1 : P (Xk = i) = p1i > 0; i = 1, ..., N ;
                          n
                          X                 n
                                            X
                                  (0)              (1)
                                 pi     =         pi     = 1.
                           i=1              i=1
                                                       m1 (0) m2          (0) mN
   Peøeíèe. LH0 (X1 , X2 , ..., Xn ) = p(0)
                                        1                p2          · · · pN      , mk  ÷èñ-
                                                                    (1) m1 (2) m2           (1) mN
ëî ñëó÷àåâ, êîãäà X = k. LH1 (X1 , X2 , ..., Xn ) =                p1     p2        · · · pN         .
Ñòàòèñòèêà íàèáîëåå ìîùíîãî êðèòåðèÿ:
                         (1)     (0)                   (1)   (0)                      (1)      (0)
ln(LH1 /LH0 ) = m1 ln(p1 /p1 ) + m2 ln(p2 /p2 ) + · · · mN ln(pN /pN ).

    Çàäà÷à 9.20. Ïóñòü X1 , X2 , ..., X15  âûáîðêà, äëÿ êîòîðîé
P (Xk = l) = pl , l = 1, 2, 3. Ïîëó÷èòü 10 ðåàëèçàöèé ýòîé âûáîðêè
        (0)                                                (1)
ñ pl = pl = 1/3, l = 1, 2, 3, è 10 ðåàëèçàöèé ñ p1 = p1 = 0, 40,
      (1)                   (1)
p2 = p2 = 0, 42, p3 = p3 = 0, 18. Äëÿ êàæäîé âûáîðêè, èñïîëü-
çóÿ íàèáîëåå ìîùíûé êðèòåðèé, íàéäåííûé â çàäà÷å 9.15, âûáðàòü
îäíó èç äâóõ ãèïîòåç. Äëÿ âû÷èñëåíèÿ îøèáîê 1-ãî è 2-ãî ðîäà α

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