Теория вероятностей и математическая статистика. Билялов Р.Ф. - 88 стр.

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S −
P
H
0
(S) = α.
P
H
0
(S
C
\S
C
S) = α − P
H
0
(S
C
S) = P
H
0
(S) − P
H
0
(S
C
S) = P
H
0
(S\S
C
S),
P
H
1
(S
C
\S
C
S) =
Z
S
C
\S
C
S
L
H
1
dx ≥ C
Z
S
C
\S
C
S
L
H
0
dx = CP
H
0
(S
C
\S
C
S) =
= CP
H
0
(S\S
C
S) =
Z
S\S
C
S
L
H
0
dx ≥
Z
S\S
C
S
L
H
1
dx = P
H
1
(S
C
\S
C
S).
P
H
1
(S) = P
H
1
(S\S
C
S) + P
H
1
(S
C
S) ≤
≤ P
H
1
(S
C
\S
C
S) + P
H
1
(S
C
S) = P
H
1
(S
C
).
H
0
: p = p
0
, H
1
:
p = p
1
p
0
< p
1
. L
H
0
= p
m
0
(1 − p
0
)
n−m
L
H
1
= p
m
1
(1 −
p
1
)
n−m
, S
C
L
H
1
/L
H
0
=
µ
¡
p
1
1−p
1
¢
/
¡
p
0
1−p
0
¢
m
¡
1−p
1
1−p
0
¢
n
≥ C, m ≥
¡
ln C + n ln
1−p
0
1−p
1
¢
/ ln
p
1
(1−p
0
)
p
0
(1−p
1
)
= m
α
, S
C
= {m ≥ m
α
}.
P
H
0
(S) =
X
m≥m
α
C
m
n
p
m
0
(1 − p
0
)
n−m
≈
1
√
2π
∞
Z
m
α
−np
0
√
np
0
q
0
=u
α
e
−
u
2
2
du = α,
m
α
= np
0
+
√
np
0
q
0
u
α
.
H
0
p =
1
2
, H
1
p >
1
2
, α = 0.05.
u
α
=
1.65, m
α
= 2020 + 1.65
q
4040
4
≈ 2072. m = 2048 ∈
¯
S,
H
0
H
1
   Äîêàçàòåëüñòâî. Ïóñòü S − íåêîòîðîå êðèòè÷åñêîå ìíîæåñòâî
PH0 (S) = α. Èìååì

PH0 (SC \SC S) = α − PH0 (SC S) = PH0 (S) − PH0 (SC S) = PH0 (S\SC S),

ïîýòîìó
                      Z                         Z
PH1 (SC \SC S) =              LH1 dx ≥ C                 LH0 dx = CPH0 (SC \SC S) =
                   SC \SC S                SC \SC S
                              Z                     Z
  = CPH0 (S\SC S) =                LH0 dx ≥              LH1 dx = PH1 (SC \SC S).
                          S\SC S                S\SC S

 èòîãå ïîëó÷àåì

                 PH1 (S) = PH1 (S\SC S) + PH1 (SC S) ≤

               ≤ PH1 (SC \SC S) + PH1 (SC S) = PH1 (SC ).
    Ïðèìåð. Áåðíóëëèåâñêàÿ ìîäåëü. Ïóñòü H0 : p = p0 , H1 :
p = p1 è p0 < p1 . Òàê êàê LH0 = pm         0 (1 − p0 )
                                                       n−m è L      m
                                                              H1 = p1 (1 −
p1 )n−m , òî êðèòè÷åñêîå  µ ìíîæåñòâî S     ¶C çàäàåòñÿ ñ ïîìîùüþ íåðà-
                              ¡ p1 ¢ ¡ p0 ¢ m ¡ 1−p1 ¢n
âåíñòâà LH1 /LH0 =             1−p1 / 1−p0       1−p0    ≥ C, îòêóäà m ≥
¡            1−p0 ¢
 ln C + n ln 1−p1
                   / ln pp10 (1−p0)
                             (1−p1 ) = mα , çíà÷èò SC = {m ≥ mα }. Âû-
÷èñëÿåì

              X                                               Z∞
                                                  1                          u2
  PH0 (S) =          Cnm pm
                          0 (1
                                       n−m
                                  − p0 )        ≈√                      e−    2   du = α,
              m≥mα
                                                  2π
                                                          mα −np0
                                                          √
                                                           np0 q0
                                                                  =uα

                  √
îòêóäà mα = np0 + np0 q0 uα .
   Ïðèëîæåíèå ê îïûòó Áþôôîíà. H0 : p = 12 , H1 : p > 12 , α = 0.05.
Èç òàáëèö äëÿ íîðìàëüíî ñòàíäàðòíîãî
                              q      ðàñïðåäåëåíèÿ íàõîäèì uα =
1.65, çíà÷èò mα = 2020 + 1.65 4040
                                 4   ≈ 2072. Íî m = 2048 ∈ S̄,
ïîýòîìó ãèïîòåçà H0 ïðèíèìàåòñÿ, à ãèïîòåçà H1 îòâåðãàåòñÿ.


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