GAUSSOVA KRIVULJAuc.fmf.uni-lj.si/mi/arhivpoletih/gradiva/1920/mr_g.pdf · 2019-09-25 · GAUSSOVA...

GAUSSOVA KRIVULJA

Martin Raic

Izzivi poucevanja matematikeUL FMF

21. september 2019

Martin Raic GAUSSOVA KRIVULJA

GAUSSOVA KRIVULJA


Carl Friedrich Gauß(1777–1855)

nemski matematik

Vir: Vikipedija


GAUSSOVA KRIVULJA (OSNOVNA)

-3 -2 -1 0 1 2 3

01

23

y = e−x2

x

y


GAUSSOVA KRIVULJA (TRANSFORMIRANA)

-3 -2 -1 0 1 2 3

01

23

y = a e−b(x−µ)2

x

y

a=1, b=1, µ=0 a=1, b=1,µ=2

a=2, b=4, µ=0

a= 12 , b=

14 , µ=0


TELESNA VISINA 1382 BAVARSKIH VOJASKIHOBVEZNIKOV IZ 19. STOLETJA

visina

frekv

enca

150 160 170 180

050

100

150

Vir podatkov (9. 8. 2015): http://www.uni-tuebingen.de/fakultaeten/wirtschafts-und-sozialwissenschaftliche-fakultaet/faecher/wirtschaftswissenschaft/lehrstuehle/volkswirtschaftslehre/wirtschaftsgeschichte/data-hub-height.html


OBSEG PRSNEGA KOSA PRI 5738 SKOTSKIHVOJAKIH

velikost

frekv

enca

35 40 45

020

040

060

080

010

00

Vir podatkov (9. 8. 2015): https://vincentarelbundock.github.io/Rdatasets/datasets.html


ZRACNI TLAK PO SVETU (761286) MERITEV

tlak [mb]

frekv

enca

9600 9800 10000 10200 10400 10600

010

000

2000

030

000

4000

0

Vir podatkov (9. 8. 2015): http://cdiac.ornl.gov/epubs/ndp/ndp026c/ndp026c.html


DEM : USD (1980–1987)

0 500 1000 1500

0.30

0.35

0.40

0.45

0.50

0.55

dan

DE

M/U

SD



DEM : USD (1980–1987)

DEM/USD

frekv

enca

0.30 0.35 0.40 0.45 0.50 0.55 0.60

050

100

150



DEM : USD (1980–1987) – SPREMEMBE

sprememba

frekv

enca

-0.02 0.00 0.02 0.04

050

100

150

200

250



TOCKE V SVETOVNEM POKALU PRI ZENSKEMSMUCANJU (2012/13)

tocke

frekv

enca

0 500 1000 1500 2000 2500

010

2030

4050

Vir podatkov (27. 3. 2013):http://www.fis-ski.com/uk/disciplines/alpine-skiing/cupstandings.html


MET 30 POSTENIH KOVANCEV: 1. izvedba

0 5 10 15 20 25 30

010

2030

4050

60

stevilo grbov

frekv

enca



0 5 10 15 20 25 30

010

2030

4050

60

stevilo grbov

frekv

enca



0 5 10 15 20 25 30

0.00

0.05

0.10

0.15

0.20

stevilo grbov

rela

tivna

frekv

enca


MET 30 POSTENIH KOVANCEV: VERJETNOSTI

0 5 10 15 20 25 30

0.00

0.05

0.10

0.15

0.20

stevilo grbov

verje

tnos

t


100 METOV POSTENEGA KOVANCA

n = 100, p = 0.5

pik

verje

tnos

t

0 20 40 60 80 100

0.00

0.02

0.04

0.06

0.08


100 METOV POSTENEGA KOVANCA

n = 100, p = 0.5

pik

verje

tnos

t

30 40 50 60 70

0.00

0.02

0.04

0.06

0.08


Abraham de Moivre(1667–1754)

francoski matematik, deloval v Angliji

Vir: Vikipedija


IZPELJAVA ZA POSTEN KOVANEC (1)

n-krat vrzemo posten kovanecX := stevilo grbov, ki padejoVerjetnost, da pade natanko g grbov, oznacimo sPn(X = k).

Pn(X = k) =

(nk

)2n(

nk

)=

n(n − 1)(n − 2) · · · (n − k + 1)

k !



Aproksimacija temelji na kvocientu zaporednih verjetnosti:

Pn(X = k)

Pn(X = k − 1)=

n!2n k! (n−k)!

n!2n (k−1)! (n−k+1)!

=(k − 1)! (n − k + 1)!

k ! (n − k)!=

=n − k + 1

k.

Zaradi enostavnosti vzemimo, da je n sodo stevilo, t. j. n = 2m.Naj bo k = m + d . Tedaj lahko zgornjo enakost prepisemo vobliki:

P2m(X = m + d)

P2m(X = m + d − 1)=

m − d + 1m + d

=1− d−1

m

1 + dm

.

V naslednjem koraku bomo to aproksimirali za primer, ko jed � m.



Za majhne x je 1 + x ≈ ex :

x

y

1 2

1

2

y = ex

y = 1 + x


Leonhard Euler(1707–1783)

svicarski matematik

Vir: Vikipedija



Ce d ni prevelik, je torej:

P2m(X = m + d)

P2m(X = m + d − 1)=

m − d + 1m + d

≈ e−(d−1)/m

ed/m = e−(2d−1)/m .

Z mnozenjem teh priblizkov dobimo (za d ≥ 0):

P2m(X = m + d)

P2m(X = m)≈ e−(1+3+5+···+(2d−1))/m = e−d2/m .

Sledi:P2m(X = m + d) ≈ P2m(X = m) e−d2/m .

Zaradi simetrije to velja tudi za negativne d .Verjetnosti torej sledijo Gaussovi krivulji.



Priblizek za P2m(X = m) dobimo iz dejstva, da je vsota vsehverjetnosti enaka 1:

m∑d=−m

P2m(X = m + d) = 1 .

Ker se vse dogaja v obmocju d = O(√

n)

(sicer je faktor e−d2/m

zanemarljiv), smemo aproksimirati:

∞∑d=−∞

P2m(X = m) e−d2/m ≈ 1 .



Prejsnjo formulo prepisemo v obliki:

√m P2m(X = m)

∞∑d=−∞

1√m

e−(d/√

m)2 ≈ 1 .

Posamezen sumand je enak ploscini pravokotnika s sirino1/√

m in visino e−(d/√

m)2, ta pa je priblizno enaka ploscini pod

krivuljo y = e−x2na intervalu okoli d/

√m s sirino 1/

√m:

x

y

d/√

m

Φ(c)

y = e−x2



Celotna vsota pa je priblizno enaka ploscini pod celotno krivuljoy = e−x2

, za katero je znano, da je enaka√π:

∞∑d=−∞

1√m

e−(d/√

m)2 ≈∫ ∞−∞

e−x2dx =

√π .

Torej je:

P2m(X = m) ≈ 1√mπ

,

P2m(X = m + d) ≈ 1√mπ

e−d2/m .



A tudi ce je n lih, se da z nekoliko tehnicno zahtevnejsoizpeljavo dobiti:

Pn

(X =

n2

+ d)≈√

2nπ

e−2d2/n

oziroma:

Pn(X = k) ≈√

2nπ

e−2(k−n/2)2/n .

Gaussovi krivulji pa sledijo tudi verjetnosti za nepostenkovanec, na katerem grb pade z verjetnostjo p ∈ (0,1), cifra paz verjetnostjo q = 1− p: za |k − np| � (npq)2/3 veljaLaplaceova lokalna formula:

Pn,p(X = k) ≈ 1√2πnpq

e−2(k−np)2/(2npq) .


Pierre-Simon de Laplace1749–1827

francoski matematik, astronom, fizik in politik

Vir: Vikipedija


100 METOV NEPOSTENEGA KOVANCA

n = 100, p = 0.1

pik

verje

tnos

t

0 20 40 60 80 100

0.00

0.02

0.04

0.06

0.08

0.10

0.12


100 METOV NEPOSTENEGA KOVANCA

n = 100, p = 0.9

pik

verje

tnos

t

0 20 40 60 80 100

0.00

0.02

0.04

0.06

0.08

0.10

0.12


CENTRALNI LIMITNI IZREK

Velja X = Y1 + Y2 + · · ·+ Yn, kjer je:

Yj =

{1 ; pri j-tem metu pade grb,0 ; pri j-tem metu pade cifra.

Gaussovi krivulji pa pod razmeroma milimi pogoji sledi tudiporazdelitev vsote veliko splosnejsih neodvisnih slucajnihspremenljivk: porazdelitve posameznih sestevancev ne smejobiti pregrde in nobeden ne sme prevec izstopati. Precizniformulaciji tega dejstva pravimo centralni limitni izrek.

Gaussovi krivulji torej sledi porazdelitev slucajne kolicine, kinastane kot rezultanta veliko majhnih neodvisnih vplivov, ki semed seboj sestevajo.


NORMALNA (GAUSSOVA) PORAZDELITEV

µ

σ σ

x

f (x)

f (x) = 1σ√

2πe−(x−µ)

2/(2σ2)

X ∼ N(µ, σ)

µ = E(X ) (pricakovana vrednost ali matematicno upanje)σ2 = var(X ) = D(X ) = E

[(X − E(X ))2]

(varianca ali disperzija)


Paul Pierre Levy(1886–1971)

francoski matematik zidovskega rodu

Vir: Vikipedija


POSTENA KOCKA

n = 1

pik

verje

tnos

t

1 2 3 4 5 6

0.00

0.05

0.10

0.15


2 META POSTENE KOCKE

n = 2

pik

verje

tnos

t

2 4 6 8 10 12

0.00

0.05

0.10

0.15


30 METOV POSTENE KOCKE

n = 30

pik

verje

tnos

t

50 100 150

0.00

0.01

0.02

0.03

0.04


GRDA PORAZDELITEV

stevilo

verje

tnos

t

0 2 4 6 8

0.0

0.1

0.2

0.3

0.4

0.5


VSOTA 10 GRDIH PORAZDELITEV

vsota

verje

tnos

t

0 20 40 60 80

0.00

0.05

0.10

0.15



vsota

verje

tnos

t

-60 -40 -20 0 20 40 60

0.00

00.

005

0.01

00.

015

0.02

00.

025

0.03

0



vsota

verje

tnos

t

-60 -40 -20 0 20 40 60

0.00

00.

005

0.01

00.

015



vsota

verje

tnos

t

-200 -150 -100 -50 0 50 100 150

0.00

00.

002

0.00

40.

006

0.00

8


DOHODKI 43886 DRUZIN V KIBERGRADU

dohodek

frekv

enca

0 50000 100000 150000 200000

050

010

0015

0020

0025

0030

0035

00

Vir podatkov: J. Rice: Mathematical Statistics and Data Analysis, Duxbury, 2007


1000 POVPRECIJ PO 20000 DOHODKOV

dohodek

frekv

enca

40800 41000 41200 41400 41600 41800

050

100

150

200



tocke

frekv

enca

0 500 1000 1500 2000 2500

010

2030

4050



LOGARITEM TOCK

log10(tocke)

frekv

enca

0 1 2 3 4

010

2030

40



STANDARDNA NORMALNA PORAZDELITEV

z

f (z)

N(0,1)

f (z) = 1√2π

e−x2/2

Z ∼ N(0,1) =⇒ a + bZ ∼ N(a, |b|)


PRIMERJALNI KVANTILNI (Q–Q) GRAFIKON:IDEJA 1

n razlicnih podatkov razdeli realno os na n + 1 delov:Primer: 3, 22, 7, −20, 11, −10, 15, 18, 25

y

−20

−10

0

10

20


PRIMERJALNI KVANTILNI (Q–Q) GRAFIKON:IDEJA 2

Realno os pa lahko razdelimo tudi glede na ploscine podGaussovo krivuljo:

x


PRIMERJALNI KVANTILNI (Q–Q) GRAFIKON:PRIMER 1

Podatki: 3, 22, 7, −20, 11, −10, 15, 18, 25

x

y

−20

−10

10

20


PRIMERJALNI KVANTILNI (Q–Q) GRAFIKON:PRIMER 2

Podatki: 5, 20, 5, −20, 10, −10, 10, 10, 25

x

y

−20

−10

10

20



velikost

frekv

enca

35 40 45

020

040

060

080

010

00




● ● ●

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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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−3 −2 −1 0 1 2 3

150

160

170

180


visi

na v

ojak

a




sprememba

frekv

enca

-0.02 0.00 0.02 0.04

050

100

150

200

250




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−3 −2 −1 0 1 2 3

−0.

020.

000.

020.

04


spre

mem

ba te

caja




tocke

frekv

enca

0 500 1000 1500 2000 2500

010

2030

4050




●

●

●

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−2 −1 0 1 2

050

010

0015

0020

0025

00


tock

e



LOGARITEM TOCK

log10(tocke)

frekv

enca

0 1 2 3 4

010

2030

40



LOGARITEM TOCK

●

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−2 −1 0 1 2

0.5

1.0

1.5

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log1

0(to

cke)



dohodek

frekv

enca

0 50000 100000 150000 200000

050

010

0015

0020

0025

0030

0035

00




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−3 −2 −1 0 1 2 3

4080

041

000

4120

041

400

4160

041

800


doho

dek


GAUSSOVA KRIVULJAuc.fmf.uni-lj.si/mi/arhivpoletih/gradiva/1920/mr_g.pdf · 2019-09-25 · GAUSSOVA...

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Transcript of GAUSSOVA KRIVULJAuc.fmf.uni-lj.si/mi/arhivpoletih/gradiva/1920/mr_g.pdf · 2019-09-25 · GAUSSOVA...