kis_marta__gazdasagi_matematika_keplettar.pdf

7/24/2019 kis_marta__gazdasagi_matematika_keplettar.pdf

1/34

GAZDASGI MATEMATIKAKPLETTR

sszelltotta:

Kis Mrta

Budapest, 2012.

orrs: http://www.doksi.hu


2/34

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



3/34

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



4/34

ANALZIS KPLETEK



5/34

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



6/34

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 =



7/34

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


8/34

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 ++=+ )(



9/34

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)(



10/34

VALSZNSGSZMTS

KPLETEK



11/34

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



12/34


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



13/34


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=



14/34


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


15/34


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)()(

==


16/34


16

NEVEZETES FOLYTONOS ELOSZLSOK

1.EXPONENCILIS ELOSZLS

1

)()( ==DM Eloszlsfggvny:

)0,(

0,1

0,0)(

>

kis_marta__gazdasagi_matematika_keplettar.pdf

Documents