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7/24/2019 kis_marta__gazdasagi_matematika_keplettar.pdf
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GAZDASGI MATEMATIKAKPLETTR
sszelltotta:
Kis Mrta
Budapest, 2012.
orrs: http://www.doksi.hu
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TARTALOMJEGYZK
I. ANALZIS KPLETEK ........................................................................................... 4
Sorozatok, sorok ......................................................................................................... 5
Pnzgyi szmtsok .................................................................................................. 5
Differencilszmts ................................................................................................... 6
Differencilszmts nhny alkalmazsa .............................................................. 7
Ktvltozs fggvnyek loklis szlsrtknek meghatrozsa ...................... 7
Integrlszmts .......................................................................................................... 8
II. VALSZNSGSZMTS KPLETEK ...................................................... 10
Kombinatorika .......................................................................................................... 11
Binomilis ttel .......................................................................................................... 11
Binomilis egytthatk tulajdonsgai ................................................................... 11Esemnyalgebra ........................................................................................................ 12
A valsznsgszmts aximi ........................................................................... 12
Valsznsgszmtsi ttelek ................................................................................ 12
Klasszikus kplet ...................................................................................................... 12
Visszatevs nlkli mintavtel ............................................................................... 12
Visszatevses mintavtel (Bernoulli-fle kplet) ................................................. 13
Feltteles valsznsg ............................................................................................ 13
Szorzsi szably ........................................................................................................ 13
Fggetlensg .............................................................................................................. 13
Teljes valsznsg ttele ........................................................................................ 13
Bayes-ttel .................................................................................................................. 14
Diszkrt eloszlsok ................................................................................................... 14
orrs: http://www.doksi.hu
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1. Binomilis eloszls ............................................................................................ 14
2. Hipergeometrikus eloszls............................................................................... 14
3. Poisson eloszls .................................................................................................. 15
4. Geometriai eloszls ........................................................................................... 15Folytonos eloszlsok ................................................................................................ 15
1. Exponencilis eloszls....................................................................................... 16
2. Normlis eloszls ............................................................................................... 16
Csebisev-egyenltlensg ......................................................................................... 17
Nagy szmok trvnye (Bernoulli-fle alak) ...................................................... 17Tbbdimenzis eloszlsok ...................................................................................... 17
III. VALSZNSGSZMTS TBLZATOK ............................................. 18
1. Binomilis egytthatk
k
n tblzata ............................................................ 19
2. Tblzat: Binomilis eloszls ............................................................................. 213. Tblzat: Poisson eloszls .................................................................................. 27
4. Tblzat: Normlis eloszls ............................................................................... 33
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ANALZIS KPLETEK
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ANALZIS KPLETEK
5
SOROZATOK,SOROK
en
n
=
+
11lim (e2,718)
1lim
en
n
=
+ , R
=
=
1
1
1n
n
q
aaq , ha ] [1;1q s 0q , Ra
PNZGYI SZMTSOKKamatos kamat
nn
n qkR
kk =
+= 00
1001 , ahol Ra kamatlb, qa kamattnyez
Diszkontls
n
nnn vk
qkk ==
1
0
qv
1= , ahol v a diszkonttnyez
( )nn dkk = 10 100/Dd= , ahol Da diszkontlb
Vsrlrtknn
nf
rk
F
Rkk
+
+=
+
+=
1
1
1001
100100 , ahol Faz inflci
Gyjtjradk
( )
1
11
=q
qqaS
n
n , ahol a az annuits
Trlesztjradk
( )
v
v
vaV
n
n
= 1
11
, vagy( )
1
11
= q
q
q
a
V
n
nn
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ANALZIS KPLETEK
6
DIFFERENCILSZMTS
Elemi fggvnyek derivltja
f f Df
c ( Rc ) 0 R
x (R) 1 R+
x
1
2
1
x
R\{0}
xe
xe
R
xa (aR+) aax ln R
xln x
1 R+
xalog (aR+\{1})ax ln
1
R+
sinx xcos Rxcos xsin R
tgx 1cos
1 22
+= xtgx
R
ctgx )1(sin
1 22
+=
xctgx
R
Derivlsi szablyok
Konstanssal szorzs: fcfc = )(
sszeg: gfgf +=+ )(
Szorzat: gfgfgf += )(
Hnyados:2
g
gfgfgf =
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ANALZIS KPLETEK
7
sszetett fggvny: ( ) ggfgf = )()(
( ) fff = 1 , R
( ) fee ff =
( ) faaa ff = ln
( ) ff
f =1
ln
( ) f
af
fa
=
ln
1log
DIFFERENCILSZMTS NHNY ALKALMAZSA
rint egyenlete: ( ) ( ) ( )000 xfxxxfy +=
Elaszticits (pontrugalmassg):( )
( )00
00
xfxf
xEx =
KTVLTOZS FGGVNYEK LOKLIS SZLSRTKNEK MEGHAT-ROZSA
I. ( ) 0, 00 = yxfx ( ) 0, 00 = yxfy
II. a) ha ( ) ( ) ( ) ( ) 0,,,, 002
000000 >= yxfyxfyxfyxD xyyyxx akkor f -nek a ( )00 ,yxP pontban loklis szlsrtke van
( )00 ,yxfxx 0 esetn minimuma;
b) ha ( ) 0, 00
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ANALZIS KPLETEK
8
INTEGRLSZMTS
Alapintegrlok
f f Df
c ( Rc ) Cxc + R
x (R) C
x+
+
+
1
1
( 1 ) R+
x
1 Cx +||ln R\{0}
xe Cex + R
xa (aR+) Ca
ax+
ln ( 1a ) R
xln Cxxx + ln R+
xalog (aR+\{1}) Cax
xx a + lnlog R+
sinx Cx + cos R
xcos C+sin R
tgx Cx + |cos|ln R
ctgx Cx +|sin|ln R
Integrlsi szablyok
HATROZATLAN INTEGRL
Cfcfc +=
Cgfgf ++=+ )(
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ANALZIS KPLETEK
9
Cf
ff ++
=+
1
1
( )1, R
Cff
f
+=
||ln
Cefe ff +=
Parcilis integrls:
= gfgfgf
HATROZOTT INTEGRL
Newton-Leibniz formula: [ ] )()()()( aFbFxFdxxf bab
a
==
ahol = )()( xfxF
=
b
a
b
afcfc
( ) +=+b
a
b
a
b
a
gfgf
+=c
a
c
b
b
a
fff
Improprius integrl
=
a
b
ab
dxxfdxxf )(lim)(
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VALSZNSGSZMTS
KPLETEK
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VALSZNSGSZMTS KPLETEK
11
KOMBINATORIKA
Permutci Kombinci Varici
Ismtlsnlkli
n!Pn = )!(!
!
knk
n
k
nC
kn
=
=
)!(
!
kn
nVkn
=
Ismtlses
!...!!
!
P
21
,...,,kn
21
r
kk
kkk
n
r
=
=
+=
k
knC ikn
1)( kikn nV =
)(
BINOMILIS TTEL
( )
),;(
...210
0
022110
RbaNnbak
n
ban
nba
nba
nba
nba
n
k
kkn
nnnnn
=
=
++
+
+
=+
=
BINOMILIS EGYTTHATK TULAJDONSGAI
=
kn
n
k
n )0;,( nkNkn
+
+=
++
1
1
1 k
n
k
n
k
n )0;,( nkNkn
n
n
nnnn2...
210=
++
+
+
)( Nn
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VALSZNSGSZMTS KPLETEK
12
ESEMNYALGEBRAAAA = AAA = ABAA = )(
HAA = =AA ABAA = )(
HHA = AHA = BABA = AA = =A BABA =
AVALSZNSGSZMTS AXIMII. 1)(0 AP
II. 1)( =HP
III. )()()( BPAPBAP += , ha =BA
VALSZNSGSZMTSI TTELEK( )APAP =1)(
( ) ( ) ( ) ( )BAPBPAPBAP +=
( )
( ) ( ) ( ) ( ) ( ) )()( CBAPCBPCAPBAPCPBPAP
CBAP
+++=
=
KLASSZIKUS KPLET
szmaeseteklehetsges
szmaesetekkedvez)( ==
n
kAP
VISSZATEVS NLKLI MINTAVTEL
=
n
N
kn
MN
k
M
pk ahol nk ,...,1,0=
ltalnostsa:
=
n
N
k
M
k
M
k
M
p r
r
kkk r
...2
2
1
1
,...,, 21 ahol nkNM
r
i
i
r
i
i == == 11
;
De-Morganazonossgok
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VALSZNSGSZMTS KPLETEK
13
VISSZATEVSES MINTAVTEL (BERNOULLI-FLE KPLET)
knkk pp
k
np
= )1( ahol nk
N
Mp ,...,1,0; ==
ltalnostsa:r
r
kr
kk
rkkk ppp
kkk
np
= ...
!...!!
!21
21 2121
,...,,
ahol nkN
Mp
r
ii
ii ==
=1
;
FELTTELES VALSZNSG( )
( )
( )BP
BAPBAP
= ahol 0)( BP
SZORZSI SZABLY
)()|()( BPBAPBAP =
Valsznsgek ltalnos szorzsi szablya:
)...|(...)|()|()(
)...(
121213121
21
=
=
nn
n
AAAAPAAAPAAPAP
AAAP
FGGETLENSG
)()()( BPAPBAP =
TELJES VALSZNSG TTELE
( ) ( ) ( ) ( ) ( ) ( ) ( )nn BPBAPBPBAPBPBAPAP +++= ...2211
rviden: ( ) ( ) ( )=
=n
kkk BPBAPAP
1
ahol nBBB ,...,, 21 teljes esemnyrendszer; nk ,...,2,1=
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VALSZNSGSZMTS KPLETEK
14
BAYES-TTEL
( ) ( )
( )
( ) ( )
( ) ( )=
=
=
n
iii
kkkk
BPBAP
BPBAP
AP
BAPABP
1
ahol nBBB ,...,, 21 teljes esemnyrendszer; 0)( >AP ; nk ,...,2,1=
VALSZNSGI VLTOZ S A VALSZNSGELOSZLS
DISZKRT ELOSZLSOK
Vrhat rtk: =
=n
iiipxM
1
)(
Szrs: ( ) ( ) 22
1
2
1
2 )( MMpxpxDn
i
n
iiiii =
=
= =
Eloszlsfggvny: RxPF
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VALSZNSGSZMTS KPLETEK
15
npM =)( 1
)(
=
N
nNnpqD
+
++= 2
1)1()mod( N
Mn ahol pq = 1 s N
Mp =
3.POISSON ELOSZLS
== ek
kPk
!)( ahol >0 s k= 0, 1, ()
=)(M =)(D
= Zha][
Zha1-s)mod(
4.GEOMETRIAI ELOSZLS
pqkP k 1)( == ahol 10 a
aFdxxfaP )(1)()(
==
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VALSZNSGSZMTS KPLETEK
16
NEVEZETES FOLYTONOS ELOSZLSOK
1.EXPONENCILIS ELOSZLS
1
)()( ==DM Eloszlsfggvny:
)0,(
0,1
0,0)(
>