close

Вход

Забыли?

вход по аккаунту

?

О приближении функции выплат смесью нормальных распределений.

код для вставкиСкачать
ISSN 2072-9502. Вестник АГТУ. Сер.: Управление, вычислительная техника и информатика. 2011. ? 1
УПРАВЛЕНИЕ
В СОЦИАЛЬНЫХ И ЭКОНОМИЧЕСКИХ СИСТЕМАХ
??? 519.87
?. ?. ???????????
О ПРИБЛИЖЕНИИ ФУНКЦИИ ВЫПЛАТ
СМЕСЬЮ НОРМАЛЬНЫХ РАСПРЕДЕЛЕНИЙ
????????
? ????????? ????? ???????? ??????? ??????? ?????????????? ?????? ????????? ??????????, ????????? ????????? ??????? ?????????? ????. ????????? ?? ????????? ????? ???????? ?????????? ??????? ????? ????????? ????????, ?? ????? ?????? ?? ??? ????? ?????? ??????????? ????????, ??? ????? ???????, ???????? ???????? ?????????? ??????.
??????? ?????????? ????? ??????? ??? ??????? ???? ? ?????? ????? [1]. ?????? ?? ???????? ????????????? ????????????? ???? ??????? ?????????? ????? ???????????? ???????????
???????? ???????? ???????? ? ??????????, ?????????? ?? ?? ??????. ? [2] ???? ???? ?????????? ??????, ??????????????? ???????????? ????????? ???????? ??? ???????????????? ??????? ????????? ????????????. ???? ?????? ???????? ???????? ? ??????? ???????????? ????????? ???????? ??????? ?????? ????????? ???????????? ? ???????? ????????????? ?????????
??? ????????????? ??????? ????????? ????????? ? ?????? ?????? ???????. ? ????????????
?????? ?????????? ??????????????? ?????? ? ??????????? ??????? ??????, ??????????
??????????? ????? ? ?????????? ??????, ?????? ?????????? ?????????????.
?????????? ??????
??? ?????????? ????????????? ???????? ??????? ?????? ???? ???? ??????? ?????????????? ?????? ?? ????? ?? ????????? ???????? (????.).
?????
??????
???????
????
??????
???
????
????
??????
162
??? ???????????
? ????????? ????????????
?????
?????? ???????????
????????????? ???????????
??????????? ???????????????
?????
?????? ???????????
????????????? ???????????
??????????? ???????????????
?????
?????? ???????????
????????????? ???????????
??????????? ???????????????
?????
?????? ???????????
????????????? ???????????
??????????? ???????????????
?????
?????? ???????????
????????????? ???????????
??????????? ???????????????
?????
?????? ???????????
????????????? ???????????
??????????? ???????????????
?????
?????? ???????????
????????????? ???????????
??????????? ???????????????
?????
?????? ???????????
????????????? ???????????
??????????? ???????????????
????????? ???????, ???.
2006 ?.
2007 ?.
36 081,48
753 232,12
1 000,00
628 220,12
35 081,48
125 012,00
0,00
0,00
46 300,58
280 852,55
26 237,05
218 153,49
19 450,68
31 889,06
612,85
30 810,00
21 943,30
196 894,69
0,00
179 688,00
21 330,45
17 206,69
612,85
0,00
53 732,60
494 615,51
53 732,60
282 210,00
0,00
212 405,51
0,00
0,00
105 916,00
206 554,26
94 680,00
124 724,26
11 236,00
81 830,00
0,00
0,00
87 529,16
150 266,00
2 712,96
91 364,00
84 816,20
58 902,00
0,00
0,00
215 216,10
298 206,17
183 157,96
78 721,07
32 058,14
219 441,69
0,00
43,41
1 762 608,72
216 342,08
1 757 848,72
55 402,78
4 760,00
160 939,30
0,00
0,00
Управление в социальных и экономических системах
??????????? ????.
?????
????????
???????
??????
???????
????????? ???????, ???.
2006 ?.
2007 ?.
11 535,54
194 713,73
2 105,54
403,00
9 430,00
194 310,73
0,00
0,00
282 237,78
310 312,70
26 734,58
3 285,89
255 503,20
307 026,81
0,00
0,00
2 614 431,90
188 263,26
2 590 417,12
33 288,00
24 014,78
154 975,26
0,00
0,00
539 431,03
168 239,21
420 455,72
297 679,00
118 975,31
465 918,21
0,00
0,00
??? ???????????
? ????????? ????????????
?????
?????? ???????????
????????????? ???????????
??????????? ???????????????
?????
?????? ???????????
????????????? ???????????
??????????? ???????????????
?????
?????? ???????????
????????????? ???????????
??????????? ???????????????
?????
?????? ???????????
????????????? ???????????
??????????? ???????????????
?????, ?? ?????? ????????? ??????, ??????? ???????? ????????????? ?????????? ??????? ????????? ?????? f1 (x ) ? f 2 ( x ) ? ?????? ??????????? ?? 2006 ? 2007 ??. ??????????????.
??? 0 < x ? 1
??? 0 < x ? 1
?1000 x,
?2007764 x + 420456,
?25237 x ? 24237,
?
??? 1 < x ? 2
?
?? 410067 x + 1038287, ??? 1 < x ? 2
?? 26237 x + 78711,
?? 38465 x + 295083,
??? 2 < x ? 3
??? 2 < x ? 3
?
?
??? 3 < x ? 4
??? 3 < x ? 4
?53733 x ? 161199,
?102522 x ? 127878,
?4947 x ? 110055,
?? 157486 x + 912154,
??? 4 < x ? 5
??? 4 < x ? 5
?
?
??? 5 < x ? 6
??? 5 < x ? 6
?? 91967 x + 554515,
?? 33360 x + 291524,
f1 ( x ) = ?
f 2 (x ) = ?
??? 6 < x ? 7
??? 6 < x ? 7
?180445 x ? 1079957,
?? 12643 x + 167222,
?1574691 x ? 10839679, ??? 7 < x ? 8
?? 23318 x + 241947,
??? 7 < x ? 8
?
?
??? 8 < x ? 9
?? 1755743 x + 15803793, ??? 8 < x ? 9
?? 55000 x + 495403,
?24629 x ? 219555,
?
??? 9 < x ? 10
2883 x ? 25544,
??? 9 < x ? 10
?
?
??? 10 < x ? 11
?2563682 x ? 25610085, ??? 10 < x ? 11
?30002 x ? 296734,
?? 2169961 x + 26459988, ??? 11 < x ? 12
?264391 x ? 2875013,
??? 11 < x ? 12
?
?
?????????? ?????? ? ??????????? ??????? f 1 (x ) ? f 2 ( x ) ?????? ?????????? ????????????? ? ??????? ??????? ???????. ??? ????? ?????? ???????
g (a, ?, x ) = g (a, ?, x, m ) = g ( x ) =
m
? pi n (ai , ?i , x ) ,
i =1
?
1
??? n (a, ?, x ) =
?e
? 2?
?m =
( x ? a )2
2 ?2
. ?????????? ????? ???????? a , ? , p ?????, ???
12
12
0
0
2
2
? ( f1(x ) ? g1 (a, ?, x)) d x + ? ( f 2 (x ) ? g 2 (a, ?, x )) d x ? min .
????????? ??????? ?????? ? ????????????? ???????????
??? ??????? ???????????? ?????? ??? ??????????? ????? MATLAB 7.5.0 (R2007b), ? ??????? ??????? ?????????? ?????????????? ??????????? ?? ??????? ??????????????? ?????.
??? ??????????? ???? ??????? ??????? lsqnonlin ?? ?????? MATLAB ? Optimization Toolbox.
163
ISSN 2072-9502. Вестник АГТУ. Сер.: Управление, вычислительная техника и информатика. 2011. ? 1
??????? lsqnonlin ???? ??????????? ??????? ?? ?????? ?????????, ??????????? ??????????? ?????? ????????????? ?????????????? ? ??????????? ?? ?????? ?????????? ?????????
???????. ?????? ???????? ???????? ? ???? ???????????? ??????? «???????» ???????? ??????? ? ??????? ?????? ?????????????? ??????????? ??????????.
??? ??????? ?????? ?????????????? ??????? ???? ??????? ?? ????????? *.m ? ?????? ?
????????? ??????? (f1.m, f2.m, g.m, ?m). ???????? ??????? ?m ????????? ??????????????
??????? ? ???????? lsqnonlin.
????????? ??????? ???????????? ?????? ?????????. ??????? ?????????? ???????????
???????? ????? ??????? ???????, ???????????? ? ???????? ?????????? ??????????? ???????
g (a, ?, x, m ) . ??? ????? ????????? ??????????????? ???????? ? m , ??? m = 1, 2, 3, ... .
??? ?????????? ???????????? ?????????? ??????? ??????? ?????????????? ?????????
???????. ???? ??? ?????????? ???????? m = m0 ???????????, ??? ? m ? ? k ??? ???? k = 1, m
? ? m +1 ? ? m , ? m + 2 ? ? m , ?? ? ???????? ???????????? ???????? m ?????????? ???????? m 0 ,
?. ?. ???????? ? m0 ???????? ?????????? ????? ?????? m 0 -? ???????? ? m , ? ??? ?? ??????,
??? ??? ????????? ?? ??? ???????? ? m . ? ?????????? ???????? ??????????, ??? ???????????
??????????? ?????????????? ??? ??????????? ??????? ??????? ???????? m = 5 .
?? ????????? ???? ????????? ??????????? ???????? a , p , ? ??? ? 5 . ? ?????????? ???? ???????? ????????? ???????? ???????? (ai , ?i , pi ) (i = 1, 5) : (0,5; 1,5; 1000000) ,
(2,9;1,2; 50000) , (5,1; 3,6; 600000) , (8; 0,7;1700000) , (11; 0,6; 2000000) .
????? ???????, ? ?????????? ?????? ????????? ???? ???????? ???????
g ( x ) = 1000000 ?
1
1,5 2?
?e
?
( x ? 0 ,5 ) 2
+ 1700000 ?
2 ?1,5
2
+ 50000 ?
1
0,7 2?
?e
?
1
1,2 2?
?e
?
( x ? 2 ,9 )2
2?1, 2
( x ? 8 )2
2? 0,7
2
+ 2000000 ?
2
+ 600000 ?
1
0,6 2?
?e
?
1
3,6 2?
?e
?
( x ? 5,1)2
2?3, 6 2
+
( x ?11)2
2 ? 0, 6 2
,
?????????? ???????????? ??????????????? ??????? f 1 (x ) ? f 2 ( x ) ?????? ?????????? ?????????????. ?? ??????????? ???? ??????? (???.) ?????????? ??????? f 1 (x ) , f 2 ( x ) ? g (x ) .
2,5 · 106
2 · 106
1,5 · 106
1 · 106
5 · 105
??????? ??????? f 1 (x ) , f 2 ( x ) ? g (x )
?? ??????? ????? ???????, ??? ?????????? ??????? ?????????? ?????? ??????????
? ??????? ??????.
164
Управление в социальных и экономических системах
??????????
?? ?????? ????????? ????????? ???? ?????????????? ?????? ???????? ?????????
? ?????? ???????? ????????????? ???????? ??????????? ??? ??????? ?????? ????????? ????????. ????? ????????????, ??? ?????????? ?????? ?? ???????? ?????????????? ??? ?????
????????? ???????? ? ?????????? ????????? ??? ??????????? ??????? ?????? ????? ????????? ???????. ?????? ???????????? ???????? ???????? ????????????? ? ????????? ??? ?????? ????????? ????????.
?????????? ??????? g ( x ) ????? ???? ???????????? ??? ?????????? ??????? ?????????
????????????? ????????? ???????? ?? ?????? ?????????????? ??????, ??????????? ? [2].
?????? ??????????
1.
2.
?????? ?. ?. ??????????? ? ????????? ???????. ? ?.: ?????, 2004. ? 96 c.
??????????? ?. ?., ????? ?. ?. ????????????? ???????? ?????? ? ????????? ???????? ?? ?????? ??????? ?????? ????????? ???????????? // ?????. ????????. ???. ????. ??-??. ? 2008. ? ? 1 (42). ? ?. 45?49.
?????? ????????? ? ???????? 19.04.2010
CONSERNING THE APPROXIMATION
OF PAYMENTS FUNCTION
BY A SET OF NORMAL DISTRIBUTIONS
O. V. Burmistrova
The increase of stability is a strategic problem of any insurance organization. Alongside with others the stability of the insurance company is also defined
by such a main factor as size of insurance payments. The optimized problem
about the approximation of payments function, occupying the central place
in mathematical models of actuarial mathematics by a set of normal distributions
is considered. Statistical data on one of the insurance companies have been assembled in order to find the approached meaning of payments function. The functions of insurance payments in personal insurance have also been made. The given optimized problem has been solved due to the application of the package
MATLAB 7.5.0. The offered technique is universal, and is applicable for various
insurance companies. The received function g(x) must be used for the numerical
analysis of various characteristics of the insurance company.
Key words: insurance organization, stability of the insurance company,
optimized problem, payments function, normal distributions, size of insurance
payments.
165
Документ
Категория
Без категории
Просмотров
5
Размер файла
162 Кб
Теги
приближение, выплате, функции, нормальной, смесь, распределение
1/--страниц
Пожаловаться на содержимое документа