ΠΑΝΕΠΙΣΤΗΜΙΟ...
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e-mail: dimitheo@aegean.gr
2010
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Pascal (1623-1662) Fermat (1601-1665)
1650 .. .
Pascal Fermat Chevalier
de Mere. .
. . On calculations in
Games of Chance Christian Huyghens (1629-1695).
, , James Bernoulli (1654-1705), Abraham de
Moivre (1667-1754), Pierre Simon Laplace (1749-1827), Simeon Denis Poisson (1781-1840) Karl Friedrich
Gauss (1777-1855). Pafnuty Chebyshev (1821-1894), Andrei Markov (1856-1922),
Richard Von Mises (1883-1953) Andrei Kolmogorov (1903-1987).
3
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.
.
.
, .
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.
.
, , ,
:
[1]. . , , : , , 1994.
[2]. S. M. Ross, A First Course in Probability, Second Edition, Macmillan Publishing Company, 1984.
[3]. P. Hoel, S. Port, C. Stone, , : A ,
, 2002.
[4]. . , , , , .
, , 2004, , 2005.
[5]. G. Roussas, A Course in Mathematical Statistics, Second Edition, Academic Press, 1997.
[6]. . & . , : , ,
, , 1999.
[7]. R. M. Spiegel, , : . , McGraw-Hill, New York,
, , 1977.
[8]. . & . , , 3 , ,
1993.
[9]. . . , , , , 1990.
[10]. . , , , , 2000.
[11]. S. M. Ross, , :
, 1 , , , 2012.
[12]. . . , , , , 2012.
4
1:
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,
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1.1
:
() k
.
()
.
.
. 1a 1v
1a 2a 2v
, ..., ,1a ...,,2a ,1ka ka kv
, ,1a ...,,2a ka
, kvvv 21 .
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3,2,1, =iTi i . 1 .
2 . 3 .
2 3222 =
5
. n
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i . 1
2 .
26 . n
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3 , .
;
. . 30235 = .
4 }.,...,,{ 21 NsssJ = J
;
. .1s
J . .,...,2 Nss
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2
;
.
. .1382400001024 43 = .6120576078910222324 =
6
1.2
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() -. .
. 1.1
1.2.
1.1 ( -). k
(i) ( )
,),1()1()(, nkknnnn kkn +== L
,nk =
.321!)(, nnn nnn L===
(ii) ( ) .kn
. (i) 1 n , 2 )1( n ,
..., k ))1(( kn .
k ( )
.)())1(()2()1( ,knknknnnn = L
(ii) 1 n , 2 n , ..., k
n .
k ( ) .knnnn = L
1.2 (- -). -
k
7
(i) ( ) knk
n,=
.)!(!
!, knk
n
k
nkn ==
(ii) ( ) ),( knN
.1
),(
+=
k
knN kn
. (i) M . k
.,kn k !k
.k - .!,
kkn
- .
k .n k
.
=
k
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.
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(ii) [1], . 48-49.
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48 .
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48
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1.3
k n .
n k .
1.3 ( ). n
k
(i) nk .
(ii) ,!!!
!,...,,;
2121 knnnn
knnn
n
L j jn ,
kjn j ,...,2,1,0 = .1
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=
. (i) ,21 knnn n ==== K
n k .
(ii)
1n
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2
1
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k
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11
11
212
1
11
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3
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nnn
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n
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=
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5 .
;
. 1.3(ii) 8=n 4=k
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!8
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=
10
6 . ()
; ()
;
. () 1.3(ii) 52=n 4=k
134321 ==== nnnn . .)!13(!52
4
() . 1T
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=
1.4
1. .
() ; () ; ()
; () ;
. () .
5
13 .
() .
1
4 .
5
13 . ,
5
13
1
4 .
() .
2
4 .
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11
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13134
1=
k kk ,
=
4
1 5
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2
4
k kk .
() . 1T
.
1
4 . 2T
. 2T
2
13 .
43 ,TT 5T .
43 ,TT 5T
1
13 .
1
13
1
13
1
13
2
13
1
4 .
2. ,)(0=
=+
n
k
knkn yxk
nyx yx, n ()
() .
. () . 1=n .1
1
0
1 0110 yxyxyxyx +=
+
=+ ,rn =
=
=+
r
k
krkr yxk
ryx
0
.)( ,1+= rn
+
=
++
+=+
1
0
11 .1
)(r
k
krkr yxk
ryx
,rn =
krkr
k
rr yxk
ryxyxyxyx
=
+
+=++=+
0
1 )())(()( 10
1
0
+
=
+
=
+
= krk
r
k
krkr
k
yxk
ryx
k
r
.210110
121112 yxr
ryx
rxy
ry
rx
r
ryx
r
ryx
rxy
r rrrrrrrr
++
+
+
+
+
++
+
= ++ KK
12
1,,2,1,1
1=
+=
+
rj
j
r
j
r
j
rK
)!1(!
)!1(
)!1(!
)1(!
)!1(!
)1(!
)!1(!
!
)!(!
!
)!1()!1(
!
1 jrj
r
jrj
rr
jrj
jrr
jrj
jr
jrj
r
jrj
r
j
r
j
r
++
=++
=++
++
=
++
=
+
.1
+=
j
r ,1
0=
=
r
rr
.1
2
1
1
1)( 11211 +++ +
+++
++
++=+ rrrrrr xyx
r
ryx
rxy
ryyx K
: +
=
++
+=+
1
0
11 .1
)(r
k
krkr yxk
ryx
,1+= rn ,
- .n
() . ),()()()( yxyxyxyx n +++=+ L n ,
nyx )( + yx, yx +
x y . x k ,
nk ,,1,0 K= y kn
.,,1,0, nkyx knk K= x
