ΠΙΘΑΝΟΤΗΤΕΣ
311
Ε ΕΙ Ι Σ ΣΑ ΑΓ ΓΩ ΩΓ ΓΗ Η ΣΤ ΤΙ Ι Σ Σ Π ΠΙ Ι Θ ΘΑ ΑΝΟ ΟΤΗΤΕΣ Σ Κ Κ Α ΑΙ Ι Τ ΤΗ Η Σ ΣΤΑ ΑΤΙ Ι Σ ΣΤΙ Κ ΚΗ (∆Ι∆ΑΚΤΙΚΕΣ ΣΗΜΕΙΩΣΕΙΣ) Χ. ∆ΑΜΙΑΝΟΥ, Ν. ΠΑΠΑ∆ΑΤΟΣ, Χ. Α. ΧΑΡΑΛΑΜΠΙ∆ΗΣ ΤΜΗΜΑ ΜΑΘΗΜΑΤΙΚΩΝ ΠΑΝΕΠΙΣΤΗΜΙΟΥ ΑΘΗΝΩΝ ΑΘΗΝΑ 2003
Transcript of ΠΙΘΑΝΟΤΗΤΕΣ
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( )
. , . , . .
2003
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1. 1 2. 1 3. 10 4. , 13 5. 18 6. 19 7. 28 8. 39 9. 43 . 1 48 2
1. H 57 2. 61 3. 65 4. 68 . 2 75 3
1. 79 2. BERNOULLI 79 3. PASCAL 85 4. 93 5. POISSON 97 . 3 103
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4
1. 107 2. ERLANG 110 3. 117 4.
POISSON
126 5. 132
. 4 135
5 ,
1. 141 2. 145 3. 147 . 5 154
1 6
1. 157 2. 158 3.
161 7
1. 179 2. 179 3. 187 4. 194 5.
196 6. 200 . 7 201
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2 8
1. 213 2. () 215 3. 218 4. 221 5. 222 . 8 227 9
1. 229 2.
231 3. 237 . 9 245 10
1. 253 2. 256 3.
265
. 10 276 283 289 297
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. , ( ) . : , , , . , : () , () , , . 1, 1-5 ( ), , (, , ) . 2-4 , 5 ( ) ( , , ), . ( ) , ( 6 7) ( 8-10). , , , . ( ) 8 10 , ( , , , , , ...), . , , ,
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vi
, . , () ( 8-10). , , . , . , ( ) , . , , , , . , , , , , . ( ) . , 2003
. , . , ..
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1 1. H ( ) ( ) ( ). . . , () ( ). , ( ). ( ) ( ) . (). 2.
() ( ) . , . (..) : ) :
..: )( , )( .
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2
)
..: 6,5,4,3,2,1 .
) )(
..: ...,,,, .
) ()
).6,6(...,),2,6(),1,6(............
),6,2(...,),2,2(),1,2(),6,1(...,),2,1(1),(1,:..
)
..: v...,,2,1,0 .
) ( )
..: ...,2,1,0 .
) .
..: .
: .
() . A , A . . BA . BA , . BA .
BA AB . BA = .
.
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3
2.1. (..) ( ) ( ) . .
, , . . . . : }...,,,{ 21 N = : ...},,{ 21 = . ( ) ( ( ) ). , :
) },{ = ) }6,5,4,3,2,1{= ) ...},,,{ = ) )}6,6(...,),2,1(),1,1{(= ) }...,,2,1,0{ = ) ...},2,1,0{= ) ),0[}0:{ +== tt . (), (), () () . () ()
...},,,{ 321 = . , () .. ),0[ = 0 ( ). ( ) , ( mm) ..,
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4
.. ( ).
2.2. . ( ). .
}{ = , , . .
, , . . () ( )
ABA = :{ }B , . , AAA ...,,, 21
jAAAA = :{21 L }...,,2,1 =j , AAA ...,,, 21 . , ...,...,,, 21 AAA
jAAAA = :{21 LL ...},2,1=j , ...,...,,, 21 AAA . () ( )
A = :{ }B , . , AAA ...,,, 21
LL 2121 jA = :{ }...,,2,1 =j ,
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5
AAA ...,,, 21 . , ...,...,,, 21 AAA
LLLL 2121 jA = :{ }...,2,1=j ,
...,...,,, 21 AAA . , = ,
( ) . ( )
}:{ AA = , . A . ( )
ABA = :{ }B , . BABA = . (). Venn ( ) , . . Venn 2.1-2.4 BA , BA , A = BA . , , . 1 2 .
1 2 , 21 , 1
2 ,
}`,:),{ 22112121 ( = .
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6
...,,, 21 :
}...,,,:)...,,,{( 22112121 = L . === L21 .
A B
2.1: BA 2.2: BA
A
2.3: A 2.4: BA 2.1. () () .
},{ = , ( ).
}{= }{= . () () 2 . 2 . 2
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7
.
)},(),,(),,(),,{(2 = . 2 },{ = . 2 ,
)},{(0 = , )},(),,{(1 = )},{(2 = 0, 1 2 , .
2.2. () .
}6,5,4,3,2,1{= .
}1{1 = , }2{2 = , }3{3 = , }4{4 = , }5{5 = }6{6 = 5,4,3,2,1 6
,
}1{1 = , }2,1{2 =B , }3,2,1{3 =B , }4,3,2,1{4 =B , }5,4,3,2,1{5 =B }6,5,4,3,2,1{6 =B
5,4,3,2,1 6
.
11 AB = , 212 AAB = , 3213 AAAB = , 43214 AAAAB = , 543215 AAAAAB = , B =6 .
2.3. 3 .
)},,(),,,(),,,(),,,(),,,(),,,(),,,(),,,{( = , . 0 , 1 , 2 3 0, 1, 2 3 , ,
:
)},,{(0 = , )},,(),,,(),,,{(1 =
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8
)},,(),,,(),,,{(2 = , )},,{(3 = ,
)},,(),,,(),,,(),,,(),,,(),,,(),,,{( =
0321 A == . () 3
0)},,{( A == . 2.4. , .
() , , ,
}...,,2,1,0{1 = . 1
}...,,2,1{ = . () , , :
}0{= . , . () , ,
}0:{2 tRt
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9
1 .
, . 2 .
2.5. () , )( )( .
, ()
...},,,,{ = . .
) 4
}{A = ) 4
...},,{ = ) 4
},,,{ = . 2.6. () , 0 1. ,
}111,110,101,011,100,010,001,000{1 = , ( .. 1 .. 2.3).
() ... 4: , , AB . ,
),(),,(),,(),,(),,(),,(),,(),,(),,(),,{(2 = , )},(),,(),,(),,(),,(),,( ,
.. ),(
.
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3.
De Moivre (1711). O , ( , ...) : , (). 1/2. . () . , ( ). Laplace (1812). , () . ( , ), () ( ). , )(AP ,
NANAP )()( = (3.1)
)(AN )(NN . )(AP
( ) (3.1)
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() : 0)( AP A , () : 1)( =P ,
() : )()()( BPAPBAP += ( ) B .
(3.1) : 0)( AN )()()( BNANBAN += , .
)()()()( 2121 APAPAPAAAP +++= LL (3.2) ( , )
AAA ...,,, 21 . (3.1) 1)( AP .
0)( =P . , = L21 AA = L21 ii A
)(/)()( iii NANAP = , ...,,2,1=i , )()()()( 21 APAPAPAP L= . (3.3)
( ) : ( ) () .
)()()(
AAP = , (3.4)
)( )( ( )
. (3.4), , (3.1). 3.1. jA j ,
.2,1,0=j )( jAP , .2,1,0=j
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12
()
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12
},{ = . , , :
21})({})({ == PP .
, 2
)},(),,(),,(),,{(2 = , },{ = . (3.3) 4 :
41
21
21})({})({)}),({( === PPP ,
41
21
21})({})({)}),({( === PPP ,
41
21
21})({})({)}),({( === PPP ,
41
21
21})({})({)}),({( === PPP .
, (3.1)
)},{(0 = , )},(),,{(1 = , )},{(2 = ,
41)( 0 =AP , 2
1)( 1 =AP , 41)( 2 =AP .
3.2. r , r < . .
=)( . , , , 2/r (. 3.1). r .
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))(( rr .
))(()( rr = .
A B
r
r/2
.
r/2
ra
3.1
, (3.4),
===r
r
rr
AP 11))((
)()()( .
, = ,
2
1)(
=rAP .
4. ,
( , ) , , NANAP /)()( = , )(AN )(NN . . ( ), , :
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. () 1
1 2 2
, 1 2
21 + . ( ). () 1
1
2 2 , 1 2
21 . ...,,, 21
().
v ...,,, 21 .
, 3=v ),,( 321 . 1 21 = (
) , 2 32 = ( ) 3 23 = ( ) , 12321 = 21, 3 :
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- }...,,,{ 21 = . - )...,,,( 21 r
r ...,,2,1= . ( ) }...,,,{ 21 r , r ...,,2,1= . . , , . ( ) . .
4.1. () , )( ,
)!(!)1()2)(1()(
=+= L , (4.1)
1 )1(321! = L ( 1)( 0 =v )1!0 =
()
,
)!(!!
!)(
==
. (4.2)
. () )...,,,( 21
}...,,,{ 21 = , 1 , , 2 ,
1 ,
1 . 121 ...,,, , , 1
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, 1)1( += . , , (4.1).
() }...,,,{ 21
! , ! . ! (4.1) (4.2).
4.2.
= L . (4.3) . )...,,,( 21
}...,,,{ 21 = i . , , (4.3).
4.3.
)!1(!)!1(
!)1()1(1
+=++=
+
L . (4.4)
. }...,,,{21 iii
}...,,,{ 21 = iii ...,,, 21 .
. iii L211 }...,,,{
21 iii }...,,,{ 21 jjj
11 ij = , ...,122 += ij , )1( += ij , 11 21 +
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4.1. () . }...,,,{ 21
}...,,,{ 21 ccc .
,
,
, . j j
j ...,,2,1= =+++ L21 ,
!!!!
21
L ,
1
1
. 2
1
2
1
.
, 1 , -
=+++ )( 121 L , , ,
11
2
1
1
LL
)!(!
)!()!(!
)!()!(!
!
1
11
212
1
11
=LLL
.
() . }...,,,{ 21 ccc .
, }...,,,{
21 iii ccc ,
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. , ( )
,
. ,
+
1
,
. 5. . ( ) , , . ( ) , , . Von Mises . ( ) A . ( )
)(n . ,
n )( .
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, Von Mises, :
nAP
)(lim)( = . (5.1)
, ,
() : 0)( AP A () : 1)( =P
() : )()()( BPAPBAP += ( ) . . (5.1) . 6. , , . .
6.1. () ( ). A () )(AP
():
() ,
0)( AP A , () ,
1)( =P , () ,
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LLL ++= )()()( 2121 APAPAAAP L++ )( AP ( ) Ai , ...,...,,2,1 i = . 6.1.
() : )()()( BPAPBAP += ( ) BA , ,
)()()()( 2121 APAPAPAAAP +++= LL , ( ) Ai ,
i ...,,2,1= . () () )(AP
A . )(AP , A . () . .
6.1. . }...,,,{ 21 =
=)( iii = }...,,,{ 21 .
)(AP
:
ii pP =})({ , Ni ...,,2,1= . , }{}{}{
21 iii A = L , ,
})({})({})({)(21 iii PPPAP +++= L
iii pppAP +++= L21)( .
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21
,
Nppp +++= L21)( ,
121 =+++ Nppp L . , , . . ,
NPp ii
1})({ == , Ni ...,,2,1= ,
)(AP
NANAP )()( = ,
.
6.2. .
}6,5,4,3,2,1{= 6)( == .
() , ( ), :
61})({ == jPp j , 6,5,4,3,2,1=j .
H
6)()( ANAP = ,
. , 5, }6,5{=A 2)( =AN ,
31)( =AP .
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() , ( ) ,
cjjPp j == })({ , 6,5,4,3,2,1=j , c . 1654321 =+++++ pppppp ,
1)654321( =+++++c 21/1=c . jjjA = }...,,,{ 21
21)( 21
jjjAP
+++= L .
5, }6,5{=A
2111
2165)( =+=AP .
6.3. ABOBA ,,,
40% 14%, 42% 4%, . , . ,
%8282.042.040.0})({}))({}),({ ==+=+= OPAPOAP . , , 18% ,
%1818.004.014.0})({})({}),({ ==+=+= ABPBPABBP .
(), () () .
6.1. () , ,
0)( =P . (6.1) () Ai , i ...,,2,1= ( )
,
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)()()()( 2121 APAAPAAAP +++= LL (6.2) () A , ,
)(1)( APAP = . (6.3) () A,B ,
)()()( ABPAPBAP = (6.4) AB ,
)()()( BPAPBAP = . (6.5) () BA, ,
)()()()( ABPBPAPBAP += (6.6)
)()()(1)( ABPBPAPBAP += . (6.7) . () =iA , ...,2,1=i , = LL AAA 21 ()
LLLL ++++== )()()()()( 2121 APAPAPAAAPP LL ++++= )()()( PPP . , () 0)( P . ,
0)()( =+++ LL PP , , 0)( =P . () =iA , ...,2,1 ++= i . () (6.1)
)()( 12121 LLL = + AAAAPAAAP )()()()()()()( 21121 APAAPAPAPAPAP +++=+++++= + LLL .
() A ( ), = AA , AA = . (6.2) 2= () 1)()()( ==+ PAPAP , (6.3).
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() BA == ABBA = :
=== ABBABABA )()()(
AABBABABA === )()()( . , (6.2) 2= ,
)()()()()]()[()( ABPBAPBAPBAPBABAPAP +=+== )()()()( ABPAPBAPBAP == . AB BAB =
)()()( BPAPBAP = . () BABA = , = BBA )( ,
BABBA = )( . (6.2), )()(])[()( BPBAPBBAPBAP +==
(6.4) (6.6). )( = BABA , (6.3) (6.7).
6.2. () AAA v ...,,, 21 . , .6.1, ,, (. 11):
i) )()()()()()()()( PBPAPBAP +++= , ii) )()()()()()()(1)( +++= .
6.1:
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6.2. AAP ),( , ]1,0[ : 1)(0 AP A (6.8)
:
)()( PAP BA , BA . (6.9) . , () ,
0)( AP , 0)( AP A (6.3), )(1)( APAP = , (6.8). , () , AB ,
0)( ABP , (6.5),
)()()( APBPABP = , BA , (6.9). 6.1 , .
6.3. . ,
)(BP .
B . )(BP )(BP . ,
8 (. 2.3) B
81)( =BP
(6.3)
87
811)(1)( === BPBP .
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)(BP
21, AA 3A
2,1 3 , .
87
81
83
83)()()()()( 321321 =++=++== APAPAPAAAPBP .
6.4. . . 365 , 366 . . , , 365 . , . )...,,,( 21 iii 365
}365,...,2,1{ , ri r ,
r ,...,2,1= . , , 365)( = .
. A . )(AP )(AP . , A
)...,,,( 21 iii 365 }365,...,2,1{ ( )
)365()( = . (6.1),
AP
365)365(
)( =
(6.3) :
APAP
365)365(
1)(1)( == .
23= , 2/15073.0)( >=P . 6.5. 10 0 9 . . 3 () .
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5. 5 BA }5,4,3,2,1,0{
}4,3,2,1,0{ . AB (6.5)
)()()( BPAPBAP = . 3
310)( =N , 10 }9...,,2,1,0{ 3 ,
36)( = , 6 }5,4,3,2,1,0{ 3 . 35)( =
091.0105
106)( 3
3
3
3== BAP .
6.6. (). 0 1 ( ).
0 1, . BA 0 1 ( ) (6.7),
)()()(1)( ABPBPAPBAP += . 39)( =AN , 9 }9...,,2,1{ 3 ,
39)( =BN , 9 }9...,,3,2,0{ 3 AB 38)( =ABN , 8 }9...,,3,2{ 3
.
054.0108
10921)( 3
3
3
3=+=BAP .
6.7. 2,1,0 3
50%, 30%, 10% 10%, . 5
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. 2,1 3.
1, 2 3. )( BAP 6.2 (ii)
)(1))(()( BAPBAPBAP == )()()()()()()(1 ABPBPAPABPPBPAP +++= .
.
()( PAP = 1 5 ) 55 )7.0())}1{(1( == P .
5)9.0()()( == PBP . )(ABP : ()( PABP = 1 2 5 )
555 )6.0()1.05.0())}3,0{(( =+== P . , 5)6.0()( =AP 5)8.0()( = . ,
()( PABP = 1 2 3 5 ) 55 )5.0())}0{(( == P .
,
5555555 )5.0()8.0()6.0()6.0()9.0()9.0()7.0(1)( +++= %29.101029.0 == . 7. , , () . , )(AP
A . A . , , ( )
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AB= | : () . , )|( ABP , ( ), , , )(AP
)(ABP .
.
7.1. 5 1 5. 1 2 3, 4 5 .
() . :
}5,4,3,2,1{1 = : }2,1{=A . , ,
52)( =AP ,
53)( =AP .
() , , . O )(BP
. , )(BP ,
. )|( ABP ,
. , )|( ABP
)(AP )(ABP ,
() , ( ) .
() , , . 20)5()( 2 == NN :
),2,3(),1,3(),5,2(),4,2(),3,2(),1,2(),5,1(),4,1{(),3,1(),2,1{(= )}4,5(),3,5(),2,5(),1,5(),5,4(),3,4(),2,4(),1,4(),5,3(),4,3( .
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To ( ), , 8)( =AN :
)}5,2(),4,2(),3,2(),1,2(),5,1(),4,1(),3,1(),2,1{(=A , o ( ), , 8)( =BN :
)}2,5(),1,5(),2,4(),1,4(),2,3(),1,3(),1,2(),2,1{(=B . , ,
52
208)()( ===
NANAP ,
() . , .
)}1,2(),2,1{(=AB 2)( =ABN .
41
82
)()()|( ===
ANABNABP .
,
NABNABP )()( = ,
NANAP )()( =
)()()|(
APABPABP = .
, , ( )
52
208)()( ===
NBNBP .
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, , , (. 7.3).
.
7.1. () ( ) A 0)( >AP . , , )|( ABP , B , :
)()()|(
APABPABP = , B . (7.1)
0)( =AP , )|( ABP . B )|( ABP .
)|( ABP , B , ,
() :
0)|( ABP B , () :
1)|( =AP , () :
LLLL ++++= )|()|()|()|( 2121 ABPABPABPABBBP ( ) ,...,...,2 ,1 , = iBi .
()
)|()|()|()|( 2121 ABPABPABPABBBP +++= LL ( ) ,...,2 ,1 , = iBi . . (7.1)
)|()()( = . (7.2) .
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7.1. ( ). Ai ,...,2 ,1=i , 0)( 121 >AAAP L .
)|()|()|()()( 12121312121 = AAAAPAAAPAAPAPAAAP LLL . (7.3) .
121221121 AAAAAAAAA LLL ,
)()()()( 121221121 APAAPAAAPAAAP LLL 0)( 121 >AAAP L ,
0)( 1 >AP , 0)(...,,0)( 12121 >> AAAPAAP L . (7.3) (). (7.1)
)()()|(
1
2112 AP
AAPAAP = , )(
)()|(
21
321213 AAP
AAAPAAAP = ,,
)()(
)|(121
121121
=
AAAP
AAAAPAAAAP L
LL
)(
)()(
)()(
)()()(121
21
21
321
1
21121
=
AAAP
AAAPAAP
AAAPAPAAPAPAAAP L
LLL
)|()|()|()( 121213121 = AAAAPAAAPAAPAP LL .
7.2. 1 r . . . jA j
...,,2,1=j . AAA L21 , (7.3),
)|()|()()( 12112121 = AAAAPAAPAPAAAP LLL
)()(
11
11
r
r
r
r =+
+= L .
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Lotto 49= 6= . r . 6=r , 6
810700000007.013998816
1 ==p .
. :
}...,,,{ 21 AAA Ai , i ...,,2,1= , , = ji AA , ji , ,
AAA = L21 , . 7.2. ( , ...).
}...,,,{ 21 AAA 0)( >AP , ...,,2,1= ,
=
=
1
)|()()( ABPAPBP . (7.4)
.
BABABABAAAB === LL 2121 )( , BA = , ...,,2,1= ji , == BAA jiji )( (. 7.2).
BA1 BA2 BA3 BAvL
1A 2A 3A L vA
7.2
, ,
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)()()()( 21 BAPBAPBAPBP +++= L . 0)( >AP , (7.2),
)|()()( ABPAPBAP = , ...,,2,1= ,
)|()()|()()|()()( 2211 ABPAPABPAPABPAPBP +++= L . 7.3. ( () Bayes). A }...,,,{ 21 AAA
0)( >AP , ...,,2,1= 0)( >BP ,
=
=
rrr
ABPAP
ABPAPBAP
1)|()(
)|()()|( , ...,,2,1=r . (7.5)
.
=
==
rrrr
ABPAP
ABPAPBP
BAPBAP
1)|()(
)|()()(
)()|( , ...,,2,1=r .
7.1. ) )( AP , ...,,2,1= , , (a priori) , )|( BAP r ,
, (a posteriori) .
) (...) Bayes 2=v , AA =1 AA =2 , 1)(0 BP
)|())(1()|()()|()()|(
ABPAPABPAPABPAPBAP += , )|(1)|( BAPBAP = .
) 2
vAAA ...,,, 21 , (7.4) (7.5)
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, vAAB L1 (. )...,,1 vAA .
) (7.4) (7.5) ...,...,,, 21 vAAA ( . =v ). 7.3. 25 . 3 2 . .
() ,
253)( =AP ,
253)( =BP .
() ,
253)( =AP
... :
253
243
2522
242
253)|()()|()()( =+=+= ABPAPABPAPBP .
7.4. 5% . 30% 1% , . () , () () . , .
05.0)( =AP , 95.0)(1)( == APAP , 30.0)|( =ABP , 01.0)|( =ABP . () :
0150.030.005.0)|()()( === ABPAPABP .
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() ... :
0245.001.095.030.005.0)|()()|()()( =+=+= ABPAPABPAPBP . () Bayes :
6122.001.095.030.005.0
30.005.0)|()()|()(
)|()()|( =+=+= ABPAPABPAP
ABPAPBAP .
7.5. , AIDS, : , : : , 25%, 25% 50%, . 5% , 1% 1. () AIDS; () AIDS, ; . () 05.0)|( =A , 01.0)|( = 001.0)|( = , ...
)()|()()|()()|()( ++= 50.0001.025.001.025.005.0 ++= %55.10155.00005.00025.00125.0 ==++= . 1.55% . ) , )|( ,
Bayes:
3125
0155.00125.0
0155.025.005.0
)()()|()|( ====
.
,
316)|(1)|( == .
(prevalence) (??) :
tt
""#""#=
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37
# = .
, 100 10 .. 4 ()
%4100/4 = %10100/10 = . . . , . , (sensitivity) (specificity) . :
+ : : + : : .
dcba +++= :
+
+ a b ba + c d dc +
ca + db + dcba +++
() :
ca
aATP +==++ ]|[ ,
dbdATP +==
]|[ .
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38
, . , (predictive value) :
ba
aTAP +==++ ]|[
dc
dTAP +== ]|[ .
, Bayes :
)()|()()|(
)()|()|( +++++++++
+= APATPAPATPAPATPTAP
)-)(1-(1)()(
)()(+=
dcba
caAP ++++== + )( .
, . :
db
bATP +==+ ]|[
ca
cATP +==+ ]|[ .
7.6. () 5%. 80% 10% .
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39
;
+A , A
, +T ( ) T ( ), :
05.0)( =+AP , 95.005.01)(1)( === + APAP , 20.0)|(1)|(,80.0)|( === +++++ ATPATPATP , 90.0)|(1)|(,10.0)|( === ++ ATPATPATP .
Bayes,
)()|()()|()()|()|( ++++
++++++= APATPAPATP
APATPTAP
%3030.090.010.005.080.0
05.080.0 =+= .
, ( , a posteriori) 30% , a-priori ( ) 5%. 8. BA , . () , =AB , 0)|( =ABP , , () ,
BA , 1)|( =ABP , . . ,
)()|( BPABP = . . , ,
-
40
)|()()|()()( BAPBPABPAPABP == ,
)()(
)|()()()()|( AP
BPABPAP
BPABPBAP === ,
)()()( BPAPABP = . . .
8.1. () ( ) BA , .
)()()( BPAPABP = . (8.1) 8.1. , B . : () () B .
)()()( ABPAPBAP = , )(1)( BPBP = ,
)()()( BPAPABP = , :
)()()](1)[()()()()()()( BPAPBPAPBPAPAPABPAPBAP ==== . A , , A , A B , ( 10). 8.1. 3 . . . AB . :
-
41
83)( = BAP ,
43
821)(1)( === APAP ,
21)( =BP .
E (8.1) .
.
AAA 321 ,,
)()()( 2121 APAPAAP = , )()()( 3131 APAPAAP = , (8.2) )()()( 3232 APAPAAP = .
1A 2A 3A -
1A 32 AA (. 8.2).
, (8.2),
)()()]([ 321321 AAPAPAAAP = , (8.3)
)()()()( 321321 APAPAPAAAP = . (8.4) , (8.2), (8.4), (8.3),
)()()]([ 312312 AAPAPAAAP = , (8.5) )()()]([ 213213 AAPAPAAAP = . (8.6)
.
8.2. () ( ) AAA ...,,, 21 . AAA ...,,, 21 ( )
)()()()(2121 iiiiii APAPAPAAAP LL = (8.7)
}...,,,{ 21 iii }...,,2,1{
...,,3,2= . , 3= (8.2) (8.4).
-
42
8.2. . 1A
, 2A
3A
-
42
.
21 , AA 3A .
366)( 2 ==N , 6 () }6...,,2,1{ 2 .
),3,4(),2,4(),1,4(),6,2(),5,2(),4,2(),3,2(),2,2(),1,2{(1 =A )}6,6(),5,6(),4,6(),3,6(),2,6(),1,6(),6,4(),5,4(),4,4( ,
)6,3(),4,3(),2,3(),6,2(),4,2(),2,2(),6,1(),4,1(),2,1{(2 =A , )}6,6(),4,6(),2,6(),6,5(),4,5(),2,5(),6,4(),4,4(),2,4( ,
)5,3(),3,3(),1,3(),6,2(),4,2(),2,2(),5,1(),3,1(),1,1{(3 =A , )}6,6(),4,6(),2,6(),5,5(),3 ,5(),1,5(),6,4(),4,4(),2,4( .
321323121 AAAAAAAAA === )}6,6(),4,6(),2,6(),6,4(),4,4(),2,4(),6,2(),4,2(),2,2{(= .
,
21
3618)()()( 321 ==== APAPAP ,
41
369)()()( 323121 ==== AAPAAPAAP ,
41
369)( 321 ==AAAP
),()()( 2121 APAPAAP = )()()( 3131 APAPAAP = , )()()( 3232 APAPAAP = ,
)()()()( 321321 APAPAPAAAP . 21, AA 3A
.
8.3. . jA j
-
43
(), 3,2,1=j . 21, AA 3A .
)},,( ),,,( ),,,( ),,,( ),,,( ),,,( ),,,( ),,,{( =
)},,( ),,,( ),,,( ),,,{(1 =A , )},,( ),,,( ),,,( ),,,{(2 =A , )},,( ),,,( ),,,( ),,,{(3 =A .
)},,( ),,,{(21 =AA , )},,( ),,,{(31 =AA , )},,( ),,,{(32 =AA , )},,{(321 =AAA .
,
21
84)()()( 321 ==== APAPAP ,
41
82)()()( 323121 ==== AAPAAPAAP ,
81)( 321 =AAAP
),()()( 2121 APAPAAP = )()()( 3131 APAPAAP = , )()()( 3232 APAPAAP = , )()()()( 321321 APAPAPAAAP = .
21, AA 3A .
9. () . 1 2 . (
) () . ( )
-
44
} , :),{( 22112121 = . . ,
=1 =2 , ,
}2 ,1 , :),{( 212 == ii .
iiA ( i ), 2 ,1=i . 21 ,
, 21 Bi , 2 ,1=i , 211 AB = 212 A =B . 1B 2B , . , =1 =2 1B 2B , . iB
i- ( i- ), 2 ,1=i . . :
1 2
)()()( 2121 BPBPBBP = (9.1) 211 AB = 212 A =B (
21 ) , . , ().
)(BP 21 B (9.1) ( ). , )( iAP iiA 2 ,1=i )(BP
21 B (9.1). ,
-
45
, .
1 2 .
)(BP 21 B )( iAP iiA 2 ,1=i , , : , (9.1), )},{( 21 21 :
})({})({)}),({( 2121 PPP = . )(BP 21 B , , ,
=B
PBP),(
2121
)}),({()(
.
211 AB = 212 A =B , )()( 11 APBP = , )()( 22 APBP = .
)()()( 2121 APAPAAP =
)()()( 2121 BPBPBBP = . ,
, ( ).
9.1. 5 . 2 . , ,
}6,...,2 ,1 ,6,...,2 ,1 :),{( === jiji ,
-
46
366)( 2 ==N . ,
}5,...,2 ,1 ,6,...,2 ,1 :),{( =++== iiijjiA , 15)( =AN . A , },1 {= .
125
3615})({ === Pp ,
127
3621})({ === Pq .
, 5
},{:),,,,{( 543215 = i , }5,4,3,2,1=i . To 5 ():
== iB :),,,,{( 54321 }}5,4,3,2,1{i
k5
,
5 . , 5 ,
})({})({})({})({})({)}),,,,({( 5432154321 PPPPPP =
=5
127
125 .
)(BPp =
=
5
127
1255p , 5...,,1,0= .
2 , 2Q ,
2 ,
6913.02412.00675.01127
1255
12711
45
102 ==
== ppQ .
9.2. N Mendel. .
-
47
, , . , ( ). , , . . , p, 2q r 12 =++ rqp .
, : 21, AA 3A
, , 21 , BB 3B
, , . ( ) , . , ,
qpqpAAPAPAAPAPAP +=+=+=2121)|()()|()()( 2211
rqqpAPAP +=+== )(1)(1)( 12 =++ rqp .
qpBP +=)( , rqBP +=)( . 21 , 3
( ) , , . AB =1 , BABA =2
BA =3 . , B A A B (. 8.1).
21 )()()()()( qpBPAPABPP +===
-
48
)()()()( 2 BAPBAPBABAPP +== ))((2)()()()( rqqpBPAPBPAP ++=+= ,
23 )()()()()( rqBPAPBAPP +=== .
. 1 1. , ,
. . . BABA ,, BA . 2. 4 : A , ( ) . .
3. ,
)(4)(3)(2 ABPBPAP == , 2/1)( = ABP . )(AP , )(BP )(ABP
)( BAP , )( BAP , )( BAP , )( BABAP . 4. 4/3)( =AP , 3/2)( =BP 5/3)( =ABP : )( BAP , )( BAP )( BAP . 5. A
5/1)(2)( += APAP )(AP .
6. 3
)()(2)( BPAPABP += , )()( BPAP = .
-
49
7. 2/1)( =AP , 3/1)( =BP 3/2)( = BAP . 2/1)( =AP , 5/1)( =BP
5/3)( =BAP . 8. 4/3)( = 8/3)( = , :
() 4/3)( ,
() 83)(
81 .
3/1)( = , 4/1)( = . 9. De Morgan:
() vv AAAA = LL 11 )( , () vv AAAA = LL 11 )( .
10. , () B , () A , () A B .
11. (i) (ii) 6.2 6.1.
12. . ( ) , 6,5,4,3,2,1= .
13. 10 . , 10...,,1,0= ;
14. ( de Mr). , 4 , 24 ;
15. . .
16. A . - . () , () rrr ...,,, 21 ...,,2,1 , , rrr =+++ L21 . 17. . r
-
50
18. }...,,,{ 21 = . })({2})({ 1+= ii PP , 1...,,2,1 = i , })({ iP , ...,,2,1=i . }...,,,{ 21 = , . 19. 5 . )6,5( , )5,6(
)6,6( .
20. 0.001. 0.97. ;
21. , , 80% , . , , 60% , 99% . . 1% , 30% . ;
22. : , : : 50 . : 30.0)( =AP , 25.0)( =BP , 40.0)( =P , 15.0)( =ABP ,
20.0)( =P , 10.0)( =P 05.0)( =P . )( BAP , )( BAP , )( AP )( BAP .
23. },...,,,{ 121 .
-
51
1 }...,,,{ 121 . 24. 10 . 100/ , 10...,,2,1= . 55/
10...,,2,1= . , 10...,,2,1= ; 25. 4/5. 3/5 2/5 .
26. : , : : . ,
)()|( BPABP < , )()|( PAP > , )()|( PP = . 27. . . . 28. 0.01. 0.95. () () .
29. 0 1 2 1/4, 1/2 1/4, . , . A , 2,1,0= rB r,
6...,,1,0=r , )( 0BP , )( 1BP )|( 12 BAP .
-
52
30. , , 50%, 40% 40%, . , 35% . 25% 20%. , 15% . ;
31. 1A 1 1 , 2 2
2 . ( ) 1
( ) 2 . 2
1A . , 1A .
() ;
() , ;
32. ( ) . () ; () ,
; ( !)
33. )521( v , 4/52, v ...,,2,1= .
34. 1 . , - ( ). 1/ v ...,,1= .
35. , s ( ), 11 + s ;
36. () . () . , 95%; 95%;
-
53
37. , . , .
38. ( Bonferroni). vAA ...,,1 ,
1)()()...( 11 +++ vAPAPAAP vv L . 39. 75%
. 20% , 10%. , ; ;
40. . 2 1 , 1 2 . , ( !), ( ) ( ). () ; () ,
;
41. , p, 2q r 12 =++ rqp . ( ) , Mendel, . .
42. 7% 2% . 48% 52% .
43. 50% . 1/2. 1/3.
44. 95% , () 95%
-
54
, () 5% . )
10%, ;
) , () . , . ;
45. , ( ). , 10%, 40% 50%, . , . , , . , 1/2. . () , ; () ,
;
46. () ( ). () . ()
0)( =AP 1)( =AP . 47. ( v2 ) .
; r )2( rvr ;
48. v...,,2,1 x . ,
y x...,,1 . y
1; y 1, x 1;
49. , , :
-
55
() B . () B . () B .
50. {=A }, =B { }, {= }. BA ,, (, ,
, A, B, ), , B, .
51. 5%. 80% , 10% . , ;
52. 21 , A 3A .
() 1A 32 AA , () 2A 31 AA () 3A 21 AA .
-
2 1. ( ) () ( ) , , , . . , () ( ) . .
1.1. () . (..). )(Xx = . WZYX ,,, XXX ...,,, 21 wzyx ,,, xxx ...,,, 21 . RRX () ( ). ],( x . XRB ( ) . .
1.2. F
}))(:({)()( xXPxXPxF == ,
-
58
(..) (...) .. .
.. XF x )(xFX .
, , ]1,0[ :
1)(0 xF ,
-
59
==
=),(,2
)},(),,{(,1),(,0
)(
X
}2,1,0{=XR . F .. :
-
60
X =)( , , . F :
-
61
1.3. x ]1,0[ 1)10( = XP
]1,0[ (
0 1). ],( 21 xx 10 21 xx
12 xx , )()( 1221 xxcxXxP =< ,
c .
0)0( =
-
62
2.1. ( ) , 1, ( )
...},...,,,{ 10 xxxRX = . f x , ...,2,1,0= ,
}))(:({)()( xXPxXPxf ==== , ...,2,1,0= , (2.1) .
.. Xf x )( xf X .
, = L}{}{ 10 xxRX L}{ x , )( XRXP
1)(0
===
xXP .
, ,
0)( xf , ...,2,1,0= 0)( =xf , XRx (2.2)
=
=0
1)(
xf . (2.3)
, }...,,,{ 10 xxxRX = , (2.3)
=
=
0
1)(xf .
)()( xXPxf == , ...,2,1,0= )()( xXPxF = ,
-
63
)()( 00 xFxf = .
=xx
xfxF
)()( ,
-
64
0)( 0 == xXP )( 0xf . , (2.9), . 0>x
xxfxxXxP )()( +< . 2.1. , 1.1, . }2,1,0{=XR (). .. :
41)}],[{()0()0( ==== PXPf ,
21)}],(),,[{()1()1( ==== PXPf ,
41)}],[{()2()2( ==== PXPf .
=
=++=2
0
141
21
41)(
xxf ,
.
2.2. x ]1,0[
(. 1.3). .
1.3
-
65
2.3.
2
)(2)(
xxf = , x 0 ,
0> . )(xF ,
-
66
})(:{ yxgx . )(xgy = XR YR .
)(1 ygx = . yxg )( )(1 ygx ,
)(xgy = )(1 ygx , )(xgy =
))(()]([)( 11 ygFygXPyF XY == ,
)(xgy = ))((1)]([1)]([1)]([)( 1111 ygFygXPygXPygXPyF XY
==
-
67
xxgy +== )( , 0 , yygx /)()(1 ==
dyydg /1/)(1 = . .
-
67
3.1. )(xf X ,
XRx . Y += , 0 ,
||1)(
yfyf XY
= , YRy . (3.2)
3.1.
= 22
2)(exp
21)(
xxf X ,
-
68
})(:{ yxgx , , . .
3.2. () )(xf X , XRx )(xFX , Rx .
2XY = .
0
-
69
. . . .
4.1. () )()( xXPxf == , ...,2,1,0= . , )( ,
=
=0
)()(
xfxXE . (4.1)
() )(xf ,
-
70
=
==+++++===6
1 27
621
6654321
61)(
xxXE .
, , .
4.2. ],[ . )(XE . ..
xf
21)( = , x .
, 4.1 (),
=
====
xxdx
dxxxfXE 042
1)()(2
.
0)( = , . . .
, )()( xXPxf X == , ...,2,1,0= )(xf X ,
-
71
. (. 4.2) . , ,
2)()( = XXg , )(XE= (4.3) (4.4) .
4.2. )(XE= . , )(XVar 2X 2 ,
])[()( 22 XEXVar = . (4.5) )(XVar ,
)(XVar X = (4.6) .
(4.3) (4.4), .. )()( xXPxf == , ...,2,1,0= ,
=
=0
2 )()()(
xfxXVar ,
.. )(xf ,
= dxxfxXVar )()()( 2 . .
.
4.1. XE =)( , 2)( XVar = , .
XE +=+ )( , (4.7) )]([)]([)]()([ XhEXgEXhXgE +=+ , (4.8)
-
72
22)( XVar =+ , (4.9) 22 )()( XEXVar = . (4.10)
. )()( xXPxf == , ...,2,1,0= . (4.3)
=
=
=+=+=+=+
0 0 0)()()()()()(
XExfxfxxfxE
=
=
=+=+=+
0 0 0)()()()()()]()([)]()([
xfxhxfxgxfxhxgXhXgE
)]([)]([ XhEXgE += . , (4.4), (4.7) (4.8). 4.2 (4.7), (4.8),
])]()[[()( 2XEXVar ++=+ 222222 ])[(])([ XEE === . 2)( X , ..
)(XE= , (4.7) (4.8) )2()()2(])[()( 22222 XEXEXXEXEXVar =+==
2222 )()(2)( XEXEXE =+= . 4.1. () .
=)(XE 0)( 2 >= XVar ,
= XZ (4.11)
, (4.7),
0/])([]/)[()( === XEXEZE , (4.9),
1/)(]/)[()( 2 === XVarXVarZVar .
-
73
(4.11) .
() 0)( =XVar , c 1)( == cXP . 4.2. . (4.10) , . ,
])[( 2)2( XE= , )1()( 2 = XXX .
22 ])[()( XEXVar += . (4.12)
(4.10)
XEXXEXXEXE ==== )()()]1([])[( 222)2( . 4.3. . )(XVar .
..
61)()( === xXPxf , 6...,,2,1=x .
, (4.3)
=
=+++++==6
1
22
691
6362516941
61)(
xxXE .
(4.10) (. 4.1) 2/7)( == XE ,
1235
449
691
27
691)()(
222 ==
== XEXVar .
4.4.
xf
21)( = , x .
(4.4)
-
74
=
===
xdxx
dxxfxXE362
1)()(23
222
(4.10) (. 4.2) 0)( == XE 3/)()( 22 XEXVar == . 4.5.
2
)(2)(
xxf = , x 0 ,
0> (. 2.3). )(XE
)(XVar .
)(XE , 4.1,
=
===33
2)(2)()(0
02
32
2
x
xdxxx
dxxxfXE
.
(4.4),
=
===623
2)(2)()(2
00
2
432
222
x
xdxxx
dxxfxXE
1896)]([)()(
22222 XEXEXVar === .
4.1. XR g
XR )]([ XgE .
(i) xg )( XRx XgE )]([ . (4.13)
(ii) xg )( XRx XgE )]([ . (4.14)
(iii) xg )( XRx XgE )]([ . (4.15)
-
75
(iv) h, )]([ XhE , )()( xgxh XRx
)]([)]([ XgEXhE . (4.16) . (iv) )()( xXPxf == , ...,2,1,0= . (4.3)
)]([)()()()()]([0 0
gExfxgxfxhhE
== =
=,
)()( xgxh , K,2,1,0= , (4.16). , (4.4).
(i) (4.16) xh )( . (ii) xg )( XRx , (4.13),
XgEgE = )]([)]([ , (4.14).
(iii) (4.13) (4.14).
A . 2
1.
-
76
-
77
() c )3/10(1
-
3
1. , 2. . () () . , . . . . 2. BERNOULLI 2.1. Bernoulli . A , ),( AA ,
= AA AA = . A . },{ = . Bernoulli.
pP =})({ , qpPP === 1})({1})({ , (2.1) .
2.1. Bernoulli p ( pq = 1 ). (-) (-) Bernoulli p. ( ))(~ pbX .
-
80
, Bernoulli .
2.1. Bernoulli p
xxqpxXPxf === 1)()( , 1,0=x . (2.2)
-
81
() . p. ( )),(~ pvbX .
, .
2.2. , . yx, ,
=
=+ v
k
kvkv yxkv
yx0
)( , ...,2,1=v .
.
),(),(),()())(()( 21 yxpyxpyxpyxyxyxyx vv LL =+++=+ ,
yxyxpp ii +== ),( , vi ...,,2,1= . kvkvv yxyxx ...,,, 1 ,
vv yxy ,..., 1 , kvk yx vk ...,,1,0= . , , vyx )( +
=
=+ vk
kvkkv
v yxCyx0
,)( ,
=kvC , kvk yx .
kvk yx k
vpp ...,,1 , x ( ,
kv y). ,
=kvC , k
=kv
pp v...,,1 ,
4.1 () . 1.
.
2.3. p
xxqpx
xXPxf
=== )()( , x ...,,2,1,0= (2.5)
-
82
-
83
(2.5) 20...,,2,1= 50.0...,,10.0 ,05.0=p . 5.0>p 5.01
-
84
222 )1()()1( pqpp =+= . ,
pqpppXEXVar =+=+== 222222 )1(])[()( . 2.1. ),( 11 zy ,
),(...,),,( 22 zyzy . zy > , zy , ...,,2,1= . p
pq =1 2/1=p . . ,
,
===
21)()(
x
xXPxf , x ...,,2,1,0= .
( ( ), ( )). () 2 () 7
8= .
=
=++=
= 2
0
8 1445.01094.00312.00039.0)5.0(8
)2(x x
XP ,
=
=+=
= 8
7
8 0351.00039.00312.0)5.0(8
)7(x x
XP .
2.2. AAA, qp 2 , r ( 12 =++ rqp ),
. ( ) , Mendel, . ( ) AA . 1
AA , Bernoulli (. 9.2 . 1)
-
85
211 )()(})({ qpPPp +=== , 211 )(1)(})({ qpPPq +=== .
Bernoulli AA
xxqpx
xf
= 11)( , x ...,,1,0= .
4/1=== rqp , 4/11 =p , 4/31 =q , AA , 4= ,
xx
xxf
=
4
43
414)( , 4,3,2,1,0=x .
4 AA
6836.0256175
431)0(1)1(
4
==
=== XPXP .
AA
1414)( === XE .
3. PASCAL 3.1. 3.1. Bernoulli p ( q),
pP =})({ , pqP == 1})({ , () . . p. ( ))(~ pGX .
.
3.1. p
-
86
1)()( === xpqxXPxf , ...,2,1=x (3.1)
-
87
.
=
= ===
1 1
11)(x x
xx xqpxpqXE
=
= ===
2 2
2122)2( )1()(])[(
x x
xx qxxpqpqxXE .
, q
=
=0
1)1(x
x qq ,
=
=1
21 )1(x
x qxq , =
=2
32 )1(2)1(x
x qqxx .
pqpxqpXE
x
x 1)1(
)(1
21
= ==== ,
22
32
2)2(2
)1(2)1(])[(
pq
qpqqxxpqXE
x
x ====
= ,
222
22 112])[()(
pq
pppqXEXVar =+=+== .
. .
3.3. (3.1).
)()|( rXPXrXP >=>+> , ...,2,1 ,0, =r . (3.4) . })(:{ rX +> })(:{ X > , })(:{ rX +>
})(:{ X > (3.2),
-
88
)()(
)(),()|(
XPrXP
XPXrXPXrXP >
+>=>>+>=>+>
rr
qq
qF
rF ==+=
+
)(1)(1
rqrFrXP ==> )(1)( (3.4).
: r ( ) r . () .
3.1. 1= XY (3.1) :
yY pqyXPyYPyf =+==== )1()()( , ...,2,1,0=y . (3.5)
.. p. H (3.3):
pqXEXEYE === 1)()1()( , 2)()1()( p
qXVarXVarYVar === . (3.6)
3.1. 100 . , 20 .
5/4=p . () 4 () .
()
1
51
54)()(
===x
xXPxf , ...,2,1=x
-
89
-
90
-
91
prXE == )( , 22 )( p
rqXVar == . (3.10)
. ..
=
==
rx
rxr qprx
xXE11
)( ,
,
===
rxx
rrxr
xrrxr
xxrx
x)!(!
!)!()!1(
)!1(11
(3.9),
prqrpq
yyr
rpqrx
xrp rr
rx
y
y
rrxr ==
+=
=
=
= 1
0)1( .
..
=
+=+=
rx
rxr qprx
xxXXE11
)1()]1([]2[ ,
,
++=+
++=+=
+
rxx
rrrxr
xrrrxr
xxxrx
xx1
)1()!()!1(
)!1()1()!()!1(
)!1()1(11
)1(
(3.9),
=
=
+++=
++=+=
rx y
yrrxr qyyr
prrqrx
xprrXXE
0]2[
1)1(
1)1()]1([
22 )1()1()1( +=+= prrqprr rr .
..
22
2
222 )1()]1([)(
prq
pr
pr
prrXXEXVar =+=+== .
3.2. r- Bernoulli p. rXY = (3.7) .
-
92
yrY qpy
yryrXPyYPyf
+=+==== 1)()()( , ...,2,1,0=y . (3.11)
.. Pascal r p. (3.10) :
prqr
prrXEYE ==== )()( , 22 )()( p
rqXVarYVar === . (3.12)
3.3. Pascal .
prX , r-
Bernoulli p ( ),(~, prNBX pr ), pY ,
Bernoulli p ( ),(~, prbY p ).
)()( ,, rYPXP ppr = , ,,...,2,1 r = (3.13) r- r.
)1()1( ,, ==+= rYpPXP ppr , ,1,...,2,1 += r (3.14) r- 1+ 1r 1+ . (3.14) Pascal.
3.3. . 49.0=p . () 4 () .
() . .. Pascal ,2=r 49.0=p
=
=++==4
2
2222 67.0})51.0(3)51.0(21{)49.0()51.0()49.0)(1()4(
XP .
-
93
() , (3.10),
08.449.02)( === XE .
3.4. Banach. Banach, o Steinhaus Pascal. . Bernoulli
2/1== qp . . . z ...,,2,1,0= . )()( zZPzf Z == , z ...,,2,1,0= .
. z
)1( + 12)()1( +=++= zzx . . , (3.7),
z
Z z
zXPzZPzf
=+====
2
212)12(2)()( , z ...,,2,1,0= .
4. , , . , . . .
4.1. , , . () o
-
94
. .. ,
. ( )),,(~ .
.
4.1. ,
+
===
x
x
xXPxf )()( , x ...,,2,1 ,0= . (4.1)
.
+=
N )(
, - . . }{ xX =
x
x
- x x .
, ,
+
===
x
x
xXPxf )()( , x ...,,2,1 ,0= .
0)( xf , x ...,,2,1,0= , 0)( =xf , }...,,2,1,0{ x Cauchy,
=
+=
x
x
x
0, (4.2)
= =
=
+
=
x
x
x
x
xf0 0
1)( ,
.
x 0 , x 0 , x 0 x
},min{},0max{ x .
-
95
.
4.2. (4.1).
XE +== )( , 1)(
2
++++==
XVar . (4.3)
. .. , ,
=
+
==
x
x
x
xXE1
)( .
=
==
11
)!()!1()!1(
)!(!!
x
xx
xx
xx
x
(4.2) Cauchy,
+
=
+
=
==
y
y
x
x
y
x
1
01 11
11
+=
+
+=
11
.
H ..
=
+
==
x
x
x
xxXXE2
)2( )1()]1([ .
=
==
22
)1()!()!2(
)!2()1()!(!
!)1()1(x
xx
xx
xxx
xx
(4.2) Cauchy
=
=
+
=
+
=
x
y
y
y
x
x
2
2
0)2( 2
2)2(
22
)1(
)1)((
)1()1(2
2)1( ++
=
+
+=
.
-
96
222
)1)(()1()1()]1([()(
+++++=+==
XXEXVar
1+
+++=
.
, N += .
4.3.
(4.1) N += . N ,, pN
N=lim ,
xx
pp
x
x
x
=
+
)1(lim , x ...,,2,1,0= . (4.4)
. pN
N=lim N
N = 1
pN
N
NN== 1lim1lim .
0lim = Nc
N ( N) c.
x
Nxx
Np
Nx
N
NN
N
N =
=
11lim)(lim L ,
x
Nxx
Np
Nx
N
NN
N
N
=
= )1(11lim)(lim L ,
111111lim)(lim =
= N
NNN
N
NL .
xxv
xx
xvx
NN
NM
N
x
M
x
x
x
)()()(
)()()(
=+
=
+
(4.4).
4.2. (Feller, 1968). A N . . .
-
97
N Np ,
.
() (4.3) Np ,
=
N
N
p N, .
N
)/()(1)/(1
)())((
,1
,
NN
NNNN
pp
N
N
=+
=
1 )/()()/( NN < 1 )/()()/( NN > . Np , N
/N < , /N > ][ /N = , ][x x. Np ,
.
5. POISSON 5.1.
!)(
xexf
x= , ...,2,1,0=x , (5.1)
xf , ...,2,1,0=x , 0)( =xf , ...},2,1,0{x ze ,
=
=0 !x
xz
xze , (5.2)
-
98
=
= ===
0 01
!)(
x x
x
eexexf ,
.
..
-
99
( 0p ) Poisson . .
Poisson .
5.2. Poisson (5.1).
XE == )( , XVar == )(2 . (5.5) . .. , ,
=
=
=== 1 11
)!1(!)(
x x
xx
xe
xxeXE ,
, (5.2) (5.5). ..
=
=
=== 22
2
2)2( )!2(!
)1()]1([
xe
xexxXXE
x
x
x
, (5.2)
2)2( )]1([ XXE == .
XXEXVar =+=+== 2222 )]1([)( . 5.1. () Bernoulli. 01.0=p . 100 .
100 .
xx
xxXP
== 100)99.0()01.0(100)( , 100...,,2,1,0=x .
-
100
100= 01.0=p 1== p 10, Poisson
!/)( 1 xexXP == , ...,2,1,0=x .
7358.03679.022)1()0()1( 1 ===+== eXPXPXP . ,
7357.03697.03660.0)1()0()1( =+==+== XPXPXP . 5.2. () Poisson. () (, ). Petri ( ) ( ). . , , . tX
t. t, tX
...,2,1,0 , t , tX ,
0t , ( ). tX
],0( t t/t = . () t/tp = , 0> , () pq =1 . tX (
)
-
101
xxt qpx
xXP
= )( , ...,2,1,0=x ,
tp .
0t , tp =lim ,
!)()(
xtexXP
xt
t== ...,2,1,0=x , )0,0( >> t . (5.6)
)(~ tPX t .
Poisson.
() . t Poisson. Rutherford, Chadwick Ellis (1920) 2608= 5.7 . Poisson 87.3= . () , . Poisson. Poisson.
() (, , ...) Poisson. .
() . .
() . t Poisson.
-
102
.
() Petri , , . t Poisson. , , Poisson.
Poisson.
5.2. 80 4 . () () ;
Poisson
!4)( 4x
exXPx
== , ...,2,1,0=x .
, ()
9084.00733.00183.01)1()0(1)2( ===== XPXPXP , . ()
=
====4
03711.01954.01954.01465.00733.00183.01)(1)5(
xxXPXP
.
5.3. 3 . : () 2 , () 4 2 , () 2 2 .
tX t
Poisson
!)3()( 3
xtexXP
xt
t== , ...,2,1,0=x .
-
103
, ()
=
=++==2
0
31 4232.02240.01494.00498.0!
3)2(x
x
xeXP
()
=
=++++==4
0
62 2851.01339.00892.00446.00149.00025.0!
6)4(x
x
xeXP .
2 ),( pb 3= 4232.0=p ( ()),
yy
yyYP
== 3)5768.0()4232.0(3)( , 3,2,1,0=y
()
3857.0)4232.0(33
)5768.0()4232.0(23
)2( 32 =
+
=YP .
. 3 1. 12 . .
2. 10 5 4 . 5 .
3. 3.0=p . 0.9.
4. r
. p . . r .
r .
r , , r
-
104
. , r . . r . . () r . )(XE
)(XVar . ()
r . )(YE
)(XVar . ()
5= , 3=r 1.0=p )(YE
15. )(YVar .
5. () a a
-
105
() .
7. Bernoulli p. () () r .
8. 125 50 . 5 . 5 2 , () 5 () .
9. 1 , . . ()
)()( xXPxf == () )(XE )(XVar . 10. 350 42 . () x () 10 3 .
11. 0.1% . 5000 () 3 () 2 () 4 .
12. Poisson. () () .
-
4 1. H () () ( ) . , ] ,[ , < .
)()( 1221 xxcxXxP =< , xx 21 , (1.1) c . x =1 , x =2 1)()( ==< XPXP
c =
1 . (1.2)
, , 0)( == xXP Rx , . . , (1.1) (1.2),
-
108
> FFXPXP
31
321 == .
() (1.5)
0)( == XE , 12/10)( 82 == XVar . 1.2. 10 , 5 .. 7:20 7:40 () 4 () 7 . , 7:20. ..
]20,0[
-
110
103)}10()13({)}0()3({)1310()30()( =+=
-
111
=== 0 220 222 1)()( dyeydxexdxxfxXE yx
=++=+== 0 020 0 0222 2]22[2][ yyyyyyy eyeeydyyeeydeydyey
22 2)(
XE = .
222 1)()(
XEXVar == .
. .
2.2. (2.1).
)()|( yXPxXyxXP >=>+> , 0x , 0y . (2.4) . }{ yxX +> }{ xX > , }{}{ xXyxX >+> (2.2),
)()(
)(),()|(
xXPyxXP
xXPxXyxXPxXyxXP >
+>=>>+>=>+>
yxyx
ee
exF
yxF
+==
+=)(
)(1)(1
yeyFyXP ==> )(1)( (2.4).
2.1. Poisson tX , 0t , tXE t =)( (. 5.2 . 3)
( ). }{ tT > , t,
-
112
}0{ =tX , t , (5.6) . 3,
tt eXPtTP
===> )0()( , 0t .. ,
-
113
,...2 ,1 ,)!1(0
1 === dxexI x , (2.7)
=== 0 0 11 1)!1(1
)!1()( dyey
dxex
dxxf yx
,
. ,...2 ,1 , =I , :
+ +=== 0 0 0 101 ][ dxexexdexdxexI xxxx
,...2 ,1,1 ==+ II . (2.8)
== 01 1dxeI x (2.7).
2.3. Erlang (2.6).
XE == )( , 22 )(
XVar == . (2.9)
. ..
==== 0 0)!1(1
)!1()()( dyey
dxex
dxxxfXE yx
(2.7)
== )!1(! .
+ + === 0 120 122 )!1(1
)!1()()( dyey
dxex
dxxfxXE yx
222 )1(
)!1()!1()(
XE +=+= .
-
114
..
22
2
2222 )1()()(
XEXVar =+=== .
2.2. Poisson tX , 0t , tXE t =)( (. 5.2 . 3) T
- ( ). }{ tT > , - t
}{ X t < , t , (5.6) . 3,
=
==== 1
0
1
0 !)()()()(
ttt
teXPXPtTP , 0t .
.. T
=
=1
0 !)(1)(
tetF t , 0t , (2.10)
0 ,0)(
-
115
() , , 12 () .
() tX t
Poisson tXE t =)( , 3/124/8 == . 3T Erlang
=
=2
0
3/
!)3/(1)(
tetF t .
=
==> 20
43 !
41)12(1)12(
eFTP
Poisson
7619.0)1465.00733.00183.0(1)12( 3 =++=>TP .
() 3T , (2.9),
93)( 3 == TE .
2.3. , ),1()( E , Erlang ,
),( ,
0> 0> , ),( . , X 0> 0> ( ),(~ X ), (.
(2.1) (2.6))
-
116
Erlang (, , ). )( , 0> , . ...},2,1{= , , }...,3,2,1{2/1 a (. ). : 0> ,
)()1( =+ , (2.14) ( , . (2.8)). ,
=)2/1( , (2.15) ( (2.15) Euler), ...),2/5(),2/3(),2/1(
(2.14) (2.15). ,
2/)2/1()2/1()12/1()2/3( ==+= , 4/3)2/3()2/3()12/3()2/5( ==+= , 8/15)2/5()2/5()12/5()2/7( ==+= ,
... ),( ,
Erlang ( 2.3). , (. (2.9))
XE == )( , 22 )(
XVar == . (2.16)
, 2/ = ( ) 2/1= , )2/1,2/( - (chi-square) (degrees of freedom),
2 . , -
( 2~ X ),
-
117
-
117
, -, , . 3. ANONIKH KATANOMH
, . :
(.. , , ...) ( ) .
. , .
.
, , .
De Moivre Laplace ),( pb ( ) Gauss . "" (Normal) Karl Pearson.
3.1. X
2 )0,( 2 >
-
118
),;( 2xf
.
3.1. () f ( ) x =
21),;(max 2
xf
x=
-
119
(, 2).
(1.5, 2) =0.5, 1 2.
1,0 == , , ),(~ 2NX
)1,0(N .
)1,0(N .
)1,0(N
Z. )(z )(z ,
,2
1)( 2/2zez =
-
120
2/2
)2/1()( xex = (0, 1). )(z . ,
-
121
== z z dyydyyz )()()( yt =
=== zz z dyydttdttz )()())(()( .
1)()()()()( ==+=+ z z dyydyydyyzz .
0=z 5.0)0( = . 1)1(2))1(1()1()1()1()11( === ZP , 1)2(2))2(1()2()2()2()22( === ZP , 1)3(2))3(1()3()3()3()33( === ZP , 1
%686826.01)8413.0(2)11( == ZP , %959546.01)9773.0(2)22( == ZP , (3.1) %7.999974.01)9987.0(2)33( == ZP .
),(~ 2NX .
-
122
3.4. X ),( 2N
() XZ /)( = )1,0(N .
() ,)(
=
XP
= )(XP ,
=
=
XP 1)( .
. () )(zFZ
XZ /)( =
),;()()( 2zFzXPz
XPzFZ +=+=
=
)2/(])[(2 22
21),;()()( zZZ e
zfzFzf +=+== )(2
1 2/2 ze z ==
)1,0(~ NZ .
()
=
=
Z
P
X
PXP )(
XZ /)( = )1,0(N )(z .
=
=
Z
PXP )( .
:
=
=
=
ZP
XPXP )( ,
.111)(1)(
=
=
==
ZP
XPXPXP
3.5. ,
X ),( 2N 2,
,
-
123
XVarXVarXE === )(,)(,)( 2 . . XZ /)( =
)()()( ZEZEXE +=+= )()()( 2 ZVarZVarXVar =+= .
)()( zzzg = )()( zgzg = , g ,
=== 0)()()( dzzgdzzzZE .
=== dzezdzezZEZVar zz )(21
210)()( 2/2/222
22
=+=+= 110)(][21)( 2/
2dzzzeZVar z .
22 1)(,0)( XVarXE ===+= . 3.1. X 270= 30= . 7
)2()2(30
27021030
270)210( =
-
124
(3.1)
%,68)11()||()( ==+ ZPXPXP %95)22()2||()22( ==+ ZPXPXP , %7.99)33()3||()33( ==+ ZPXPXP .
68% , 95% 99.7% .
.
3.3. )1,0(~ NZ z )( z= zZP => )( , 10
z = )(1 z =1)( . 01.0=
99.001.01)( ==z 1
33.2z .
-
125
05.0= 95.005.01)( ==z 645.1=z ,
10.0= 90.010.01)( ==z 28.1=z .
z
zZP => )( , 10 XP )1,(~ 2NX .
999.0001.01)75(1)75( ==>= XPXP
999.01
751
=
XP
999.01
75 =
.
-
126
1
09.31
75 =
09.375 = . 91.7109.375 == . 3.5. X min30=
min2.1= . 33min
=
==>2.13033
2.1301)33(1)33( XPXPXP
%6.00062.09938.01)5.2(1)5.2(1 ==== ZP . 28min
%50475.0)67.1(1)67.1(2.13028
2.130)28( ===
= ZPXPXP .
10 2 28min
Y = ( 10) 28min.
),10(~ pbY 05.0)28( == XPp )1()0(1)2(1)2( ===
-
127
, 100= .
, p ),( 2N
, p = , )1(2 pppq == . 100= .
-
128
De Moivre 1733 5.0=p p )10(
-
129
4.2. ),(~ pbX
pqp
pqpXP )( .
. . , )( kXP = , ...,1,0=k ),( pqpN k,
21k
21+k
+=
pq
pk
pq
pkkXP
2
121
)( .
4.3. ( ). ),(~ pbX ( p )
0 ,
+
pq
p
pq
pXP 2
121
)( .
p 2/1 . , , Poisson. , Poisson ( =)(XE , XVar =)( )
),( 2 = , = .
k
e
kXP 2)( 2
21)(
=
-
130
XP )( ,
+=
k
kkXP 2
121
)( ,
+
XP 2
121
)( .
4.1. 20%. 100 26 ; 100 ,
2.0=p 100= .
kk
kkXP
== 100)8.0()2.0(100)( , 100...,,1,0=k
=
= 100
26
100)8.0()2.0(100
)26(k
kk
kXP . (4.1)
. ,
=8.02.01002.010026
8.02.01002.0100)26( XPXP
0668.09332.01)5.1(1)5.1(16
2026 ====
ZPZP .
-
131
)375.1(8.02.0100
2.01002126
)26( =
ZPZPXP
0845.09154.01)375.1(1 === . )26( XP (4.1) 0875.0 .
4.2. p . p 1% 95%; 03.0p ( ) ; , p. /X ,
95.001.0
pXP .
=
=
01.001.001.0 pXPp
XP =+ ]01.001.0[ pXpP
+=
pqpp
pqpX
pqppP )01.0()01.0(
101.0201.001.0
=
pq
pq
pq ,
95.0101.02
pq
-
132
975.001.0
pq .
1
96.101.0 pq
pq 38416 . (4.2) 4/1)1( = pppq ( 2)1()( pppppg == 5.0p 5.0p 25.0)5.0()(max == gpg
p)
(4.2)
96044138416 .
03.0p 0021.0)03.01(03.0)1( == pppq ,
(4.2)
810021.038416 . 4.3. Poisson 200 .
() 170 ; () 11
170;
A 1 , Poisson 200= ),( 2N 200= , 200= . ()
=
= )12.2(200
200170200
200)170( ZPXPXP
)12.2()12.2(1)12.2(1 === ZP 983.0= . ,
-
133
)16.2(200
2005.0170200
200)170(
= ZPXPXP
%.5.989846.0)16.2( == () ( ) 170. ),(~ pbY ,
12= , 9846.0=p .
1211
1212
1112
)12()11()11( pqpYPYPYP
+
==+==
1211 )9846.0(0154.0)9846.0(12 += 9859.08301.01558.0 =+= . 5. , . Xlog .
. Xlog
( ) . Xlog
SGPT (serum glutamic pyruvic
transaminase) , Xlog .
, .
,
5.1.
(lognormal) 2 )0,( >
-
134
XY log= ),( 2N .
, )(xF ( 0>x ) :
===
x
XPxXPxXPxF loglog)log(log)()(
=
x log (5.1)
=
==
xx
x
xxFxf log1loglog)()( , 0>x .
5.1. 2
2
2
2)(log
21)(
x
ex
xf
= , 0>x .
r ( )
22
21
)(rrr eXE
+= ...,2,1=r
2
21
)(
eXE+= , 2222)( eXE +=
)1())(()1())(()()(222 2222 === + eXEeeXEXEXVar .
(5.1).
-
135
SGPT 25.
2 .. ,
2
21
)(
eXE+= , )1())(()( 22 = eXEXVar
54.182
21
=+ e , 03.14)1()54.18( 22 =e .
04.0)54.18(
03.141 22 ==e
04.004.1log2 == .
92.254.18log21 2 ==+
9.204.02192.2 == .
==2.0
9.225log2.0
9.2log)25log(log)25( XPXPXP
9452.0)6.1(2.0
9.222.3 ==
= .
%5.94 25 SGPT . . 4
1. ],[ . 1)( =XE 3)( =XVar ,
) ,
-
136
) || XY = ) )(YE )(YVar .
2. ]1,0[ . )(XgY = ,
yxg =)( 111 +
-
137
i) 140 150, ii) 130 160.
8. 295 240 . 9 10 , .
9. 250 50. )
200 260. ) c 10%
c.
10. 0.04. 02.0 (
02.0 02.0+ ). ) ; ) 20 .
i) ; ii) 2 ; iii) 3 ; iv) 6 ; ) i) 20 ; ii) 10 ;
11. K cm175= cm5= . ) i) 175 cm; ii) 180 cm;
-
138
iii) 170 cm 180 cm; ) 6 i) 180 cm; ii) 4 ;
12. X 2 c
)(2)( cXPcXP =>
c =+ 43.0 . : K 110 mg/dl
2)/5( dlmg , c
.
13. )1,0(~ NZ z zZzP = 1)( , 10
-
139
p 0.05 0.99. 80% ;
18. .
73.2)( =XE , 075.0)( =XVar . )
2.71 2.74; ) 10
2.71 2.74; ) 10 2
2.71 2.74;
19. 5.1 SGPT 64.34)( =YE 113)( =YVar . 25 , : 25X , 25>X . ;
20. (). , c, %100
. c ; %1= , %5 , %10 . ;
-
5 , 1. 1 () BA, ,
. .
1.1. () YX , () ,
)()(),( yYPxXPyYxXP = , (1.1)
x y. () , vXXX ...,,, 21 ()
)()()()....,,,( 22112211 vvvv xXPxXPxXPxXxXxXP = L (1.2)
vxxx ...,,, 21 .
1.1. () (1.1) : .. , .. 2X , , (1.1),
)()2|( yYPXyYP = , y. .. .. ( ). () (1.1) ,
)(),( BAPyYxXP = , })(:{ xA = , })(:{ yB = .
-
142
, (1.2)
})(:{ ii x , vi ...,,2,1= . () (1.2) : vBB ...,,1 ,
)()()...,,( 1111 vvvv BXPBXPBXBXP = L . () , (.. , ...).
.. . , , .
1.1. () .. vXXX ...,,, 21
vfff ...,,, 21 , ,
)()()()...,,,( 22112211 vvvv xfxfxfxXxXxXP L==== ,
vXvXX RxRxRx ...,,, 21 21 , iXR iX , vi ...,,2,1= .
() .. vXXX ...,,, 21 vfff ...,,, 21 , ,
)()()()...,,,( 221121...,,, 21 vvv xfxfxfxxxf v L= , vxxx ...,,, 21 ,
)...,,,(...
)...,,,( 221121
21...,,, 21 vvv
vXXX xXxXxXPxxxxxxf
v
=
vXXX ...,,, 21 .
: vXXX ...,,, 21 ,
vggg ...,,, 21 , ..
)(...,),(),( 222111 vvv XgYXgXg ===
-
143
. , .. )( iii XgY = .. iX ,
. , , .. ..
)...,,( 111 kXXgY = , )...,,( 112 vk XXgY += , 11 vk , vXX ...,,1 , .. 1Y 2Y
.. ( .. ),( 2111 XXgY = ),( 3122 XXgY = , .. 1X .. ), 21 YY .
..
1.2. .. vXX ...,,1 ,
(i) ][][][ 11 vv XEXEXXE LL = , ,
(ii) )]([)]([)]()([ 1111 vvvv XgEXgEXgXgE LL = , ( ).
H .
1.1. vXX ...,,1 ,
(i) )()()( 11 vv XVarXVarXXVar ++=++ LL , (ii) )]([)]([)]()([ 1111 vvvv XgVarXgVarXgXgVar ++=++ LL ( ).
. )()( 11 vv XgXgY ++= L . 22 )]([)()( YEYEYVar = .
)]()([)( 11 vv XgXgEYE ++= L )]([)]([ 11 v XgEXgE ++= L
++= L1 , )]([ iii gE = , (. 4.1, (4.8) . 2),
-
144
= =
=++=v
i
v
jji YE
1 1
21
2 )()]([ L .
211
2 )]()([ vv XgXgY ++= L = =
=v
i
v
jjjii XgXg
1 1)()( ,
=
= =
v
i
v
jjjii XgXgEYE
1 1
2 )()()( = =
=v
i
v
jjjii XgXgE
1 1)]()([ .
,
22 )]([)()( YEYEYVar = ])]()([[1 1
j
v
i
v
jijjii XgXgE
= == .
1.2 (ii), ji jijjiijjii XgEXgEXgXgE == )]([)]([)]()([ ,
ji XX , . ,
{ }=
= vi
iii XgEYVar1
22]))([()( =
=v
iii XgVar
1)]([ ,
(ii). (i) (ii)
iii XXg =)( , vi ...,,2,1= . 1.1. Bernoulli vXXX ...,,, 21 ,
p ( ), . pXP i == )1( , qpXP i === 1)0( , vi ...,,2,1= . ..
vXXX ++= L1 (1.3) , , p, ),(~ pvbX . }...,,1,0{ vRX = . vp = vpq =2 . 2. (1.3)
vp =++=++== )()()()( 11 LL ( pXE i =)( , )...,,2,1 vi = . 1.1 , vXX ...,,1 pqXVar i =)( . ,
-
145
vpqXVarXVarXXVarXVar vv =++=++== )()()()( 112 LL , . 2. ,
vv XaXaY ++= L11 , vaa ...,,1 , :
= = ==
===
= vi
v
i
v
iiiiii
v
iii apXEaXaEXa
1 1 11)()()( ,
==
=
= vi
ii
v
iii XaVarXaVarYVar
11)()(
= === v
i
v
iiii apqXVara
1 1
22 )(
( .. ii Xa , vi ...,,2,1= , ). , 0)( 21 = XXE , pqXXVar 2)( 21 = .
1.2. iX i
2i , vi ...,,2,1= ( )),(~ 2iii NX ,
==
=
vi
ii
v
iii aXaE
11
===
v
iii
v
iii aXaVar
1
22
1.
,
2121 )( XXE = , 222121 )( XXVar += . 1.3. i .. Poisson, )(~ ii P ,
vi ...,,2,1= , 0>i ,
==
=
vi
ii
v
iii aXaE
11
===
v
iii
v
iii aXaVar
1
2
1
( iii XVarXE == )()( )(~ ii PX ). 2.
.. , .
2.1. vXXX ...,,, 21 ..
-
146
(i) ( Bernoulli ( )). ),(~ pvbX ii , vi ...,,2,1=
= =
= v
i
v
iii pvbXX
1 1,~ ,
, iX Bernoulli, ),1()(~ pbpbX i , ),(~1 pvbXX v++L .
(ii) ( (Pascal) ). ),(~ prNBX ii , vi ...,,2,1= ,
= =
=
v
i
v
iii prNBXX
1 1,~ ,
, iX , ),1()(~ pNBpGX i , ),(~1 pvNBXX v++L .
(iii) ( Poisson). )(~ ii PX , vi ...,,2,1= ,
= =
= v
i
v
iii PXX
1 1~ .
(iv) ( ). ),(~ aX ii , vi ...,,2,1= ,
xa
i
a
X exaxf i
i
i/1
)()( = , 0x ,
dueua uai i = 0 1)( , 0>ia ,
Euler (. 2.3 . 4),
= =
v
i
v
iii aX
1 1,~ .
, iX 0> , . ),1(),1()(~ EEX i , ),(),(~1 v ++L .
(v) ( ). ),(~ 2iii
= = =
v
i
v
i
v
iiii NX
1 1 1
2,~ ,
-
147
,
= = =
++
v
i
v
ii
v
iiiiii X
1 1
2
1
2,~ .
, ),(~ 21 NX ),(~2
2 ( ),
)2,3(~3 221 . , . 3.
, , ,
vXXS ++= L1 , v . , v , ..
vXX ...,,1 .
3.1. vXXX ...,,, 21 ..
F (, )~...,,, 21 FXXX v . vXXX ...,,, 21
. vXXX ...,,, 21
(= , . ), i.i.d.= independent, identi-cally distributed.
3.1. vXXX ...,,, 21
F. XE i =)( 2)( XVar i = ,
-
148
vS
)(1 =++= L , v XXS ++= L1)( .
.
)()( 1)( v XXESE ++= L vXEXE v =++= )()( 1 L ,
SEv
S
==
= )(11)( )()( . , ,
211)( )()()()( vXVarXVarXVarSVar vv =++=++= LL
SVar
vS
vVarXVar
2
)(
2
)( )(11)( =
=
= .
XXVar )(
)()(
2
== ,
vS
SSVarSES
== )(2
)(
)(
)()(
)()(
,
S
v
S
v
S
Xv
=
=
= )()()(
)( .
(3.1). ,
)()(
XvE =
0)())(( ===
,
=
)()(
vVar
XvVar )()( 2
2
XVarvXVar
v =
=
1)(2
22 ===
vXVar
.
1.1. ( ...,,1 ) )(S
...,,1 . 3.1
-
149
Xv
XVar )(
)()( =
vS
SVarSES
= )()(
)()(
)()(
.
, , ( , ) .
3.2. ( , ...). vXXX ...,,, 21 .. F
( ) XE i =)( , 2)( XVar i = ,
-
150
(ii) ),(~ pvbX , ( p),
=+
== + rvuu
rvu rr
r
u 22
12)1()(
)(
=++=+
=+== ++ rvxxzz
rvxzrrrr
rr
22
2][2
12)1()(1
)(
=+
=+= + rvxx
rvx rr
r
22
12)1()(
)(
+= . 0 (ii) 5.2.
) 5.1, 5.2 (;) (;)
5.1 . (coding).
1 . 0y .
2K .
cyy
u ii0= , ki ...,,2,1= .
3K . u 2us
=
= ki
iiuvvu
1
1 ,
= =k
iiiu uvuvv
s1
222
11 .
4K . x 2xs
0yucx += , 222 ux scs = .
-
199
2K ,...,2,1,0 =iu .
5.1. 2.2 ( 2.3) ( ) .
iy iv iN ii yv 2iy 2ii yv
5.5 8.5
11.5 14.5 17.5 20.5
8.5 11.5 14.5 17.5 20.5 23.5
7 10 13 16 19 22
4 16
3 4 0 1
4 20 23 27 27 28
28 160
39 64 0 22
49 100 169 256 361 484
196 1600
507 1024
0 484
28 313 3811
178.1128
313 ==x , 56.1128
3131811271 22 =
=s .
310= ii yu , 6...,,2,1=i
iy iv iu iiuv 2iu 2iiuv
7 10 13 16 19 22
4 16
3 4 0 1
-1 0 1 2 3 4
-4 0 3 8 0 4
1 0 1 4 9
16
4 0 3
16 0
16 28 11 39
3929.02811 ==u , 284.1))3929.0(2839(
271 22 ==us
-
200
178.113929.0310 =+=x , 56.11284.192 ==xs . . , Sheppard ( 3)
81.1012956.11
12
222 === css ,
10.979 (. ). 6.
( ) , ,
xsCV =
(coefficient of variation). ,
%100
==CV .
, , , . . 10%. .
6.1. 30 600 75 , 20 500 70 .
-
201
( ).
%5.12%10060075 ==ACV
%14%10050070 ==BCV .
, . Gini. Gini
xdg2
=
d Gini x .
3.2 Gini
24.06.152
44.7 ==g
3.3
11.027.242
35.5 ==g .
Gini Gini CV .
7.
1. 20 :
86.283.165.112.411.387.276.390.314.255.250.160.336.155.293.170.120.380.296.131.2
) N i) , ii) , iii) ,
-
202
iv) , v) 3 , vi) Gini Gini. ) N (stem-leaf plot)
(box-plot). ) N 5
(i)-(vii) () .
2. D.N.A. 52
1.40.30.30.138.38.27.81.39.51.45.18.29.23.29.14.18.25.34.14.30.36.56.19.37.03.34.40.27.13.39.37.28.79.25.71.36.34.24.16.34.47.29.32.139.29.38.33.14.17.62.134.3
) N :
i) , ii) , iii) , iv) , v) 1 3 , vi) Gini Gini, vii) .
) N (stem-leaf plot) (box-plot).
) 8 (i)-(vii) () .
3. 72 500 :
5228364124163131517251
1831214518139201029202721101118726
12103224142173342131115
11754156091917962473
-
203
) :
i) , ii) , iii) , iv) , v) 1 9 , vi) Gini Gini.
) (stem-leaf plot) (box-plot).
) 8 (i)-(vii) () . .
4. 267 mg/100ml :
3.0 3.4 3.5 3.9 4.0 4.4 4.5 4.9 5.0 5.4 5.5 5.9 6.0 6.4 6.5 6.9 7.0 7.4 7.5 7.9 8.0 8.4 8.5 8.9
2 15 33 40 54 47 38 16 15
3 1 3
) :
i) ( ), ii) , , iii) 1 9 , vi) .
) .
5. 4
-
204
LL734.9-4.0173.9-3.0
O
.
6. 80 ( ) .
1214.5-12.51712.5-10.52310.5-8.588.5-6.5
11 6.5-4.57 4.5-2.52 2.5-0.5
)(
%)(
) : i) ( ), ii) , , iii) 1 9 , iv) o .
) .
7. 600 :
8079-779077-75
10075-7315073-718071-696069-674067-65
) :
-
205
i) ( ), ii) , , iii) 1 9 , iv) .
) .
8. ) (box-plots) ( cm) 1963 1982
1963 1964 1965 1966 1967 1968 1969 1970 1971 1972
108 165
79 77
132 99 85
100 68
123
106 138 125 103 128 132 118 117 120 114
1973 1974 1975 1976 1977 1978 1979 1980 1981 1982
129 79
180 92
105 99
168 219 135 150
130 104 144 108 152 119 135 155 134 116
) (ogive) . ().
) (ogives) 80% .
) .
9. 1v 2v () 1x
2x 21s ,
22s .
21 vvv += , x 2s
xvxvx 2211 += 212212222112 )()1(1
