pijan'othtes kai statistik'h december 21, 2010 · mèsh tim tuqaÐwn metablht¸n e(x) =...

Pijanìthtec kaiStatistik

S. Malef�kh

M�jhma 50

S. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 21 DekembrÐou 2010

Mèsh tim tuqaÐwn metablht¸n

E (X ) =

{ ∑x xf (x) an h t.m. Q eÐnai diakrit ∫

x xf (x)dx an h t.m. Q eÐnai suneq c

'Estw Q o arijmìc rÐyhc enìc zarioÔ. BreÐte thn mèsh tim thc

Q. (E(Q) = 3.5)

BreÐte th mèsh tim thc t. m. Q thc opoÐac h sun�rthsh

puknìthtac pijanìthtac isoÔtai me

f (x) =1

100e−x/100, x > 0

(E (X ) = 100)

E (X ) =

(E(Q) = 3.5)

f (x) =1

100e−x/100, x > 0

(E (X ) = 100)

E (X ) =

Q. (E(Q) = 3.5)

f (x) =1

100e−x/100, x > 0

(E (X ) = 100)

E (X ) =

Q. (E(Q) = 3.5)

f (x) =1

100e−x/100, x > 0

(E (X ) = 100)

E (X ) =

Q. (E(Q) = 3.5)

f (x) =1

100e−x/100, x > 0

(E (X ) = 100)

Mèsh tim sun�rthshc tuqaÐwn metablht¸n

'Estw g(·) mÐa pragmatik sun�rthsh. Tìte

E (g(X )) =

{ ∑x g(x)f (x) an h t.m. Q eÐnai diakrit ∫

x g(x)f (x)dx an h t.m. Q eÐnai suneq c

Idiìthtec

E (α) = α

E [X − E (X )] = 0

E (X + β) = E (X ) + β

E (αX + β) = αE (X ) + β

An Q kai Y t.m. tìte E (X + Y ) = E (X ) + E (X )

An Q kai Y anex�rthtec t.m. tìte E (XY ) = E (X )E (Y )

E (g(X )) =

Idiìthtec

E (α) = α

E [X − E (X )] = 0

E (X + β) = E (X ) + β

E (αX + β) = αE (X ) + β

E (g(X )) =

Idiìthtec

E (α) = α

E [X − E (X )] = 0

E (X + β) = E (X ) + β

E (αX + β) = αE (X ) + β

E (g(X )) =

Idiìthtec

E (α) = α

E [X − E (X )] = 0

E (X + β) = E (X ) + β

E (αX + β) = αE (X ) + β

E (g(X )) =

Idiìthtec

E (α) = α

E [X − E (X )] = 0

E (X + β) = E (X ) + β

E (αX + β) = αE (X ) + β

E (g(X )) =

Idiìthtec

E (α) = α

E [X − E (X )] = 0

E (X + β) = E (X ) + β

E (αX + β) = αE (X ) + β

E (g(X )) =

Idiìthtec

E (α) = α

E [X − E (X )] = 0

E (X + β) = E (X ) + β

E (αX + β) = αE (X ) + β

Diaspor� tuqaÐwn metablht¸n

Var(X ) = σ2 = E((x − µ)2

{ ∑x(x − µ)2f (x) an h t.m. Q eÐnai diakrit ∫

x(x − µ)2f (x)dx an h t.m. Q eÐnai suneq c

Tupik Apìklish

σ =√σ2

Suntelest c DiakÔmanshcσ

µ× 100%

mètro metablhtìthtac thc katanom c pou den ephre�zetai

apì thc mon�dec mètrhshc

ekfr�zei thn tupik apìklish thc katanom c wc posostì

thc mèshc tim c

Var(X ) = σ2 = E((x − µ)2

Tupik Apìklish

σ =√σ2

µ× 100%

thc mèshc tim c

Var(X ) = σ2 = E((x − µ)2

Tupik Apìklish

σ =√σ2

µ× 100%

thc mèshc tim c

Var(X ) = σ2 = E((x − µ)2

Tupik Apìklish

σ =√σ2

µ× 100%

thc mèshc tim cS. Malef�kh Tm ma QhmeÐac Pijanìthtec kai Statistik 21 DekembrÐou 2010

Suntelest c Loxìthtac

α3 =E (x − µ)3

Metr�ei to bajmì kai th dieÔjunsh asummetrÐac thc katanom c

α3 = 0 summetrik katanom

Suntelest c KÔrtwshc

α4 =E (x − µ)4

EÐnai èna mètro thc aiqmhrìthtac thc katanom c sto kèntro

thc. 'Oso megalÔterh eÐnai h tim tou α4, tìso aiqmhrìterh

eÐnai h katanom .

Sundiaspor�

eÐnai èna mètro metablhtìthtac dÔo t.m. Q kai Y .

Cov(X ,Y ) = E ((X − µX )(Y − µY ))

= E (XY )− µXµY

An Q kai Y anex�rthtec t.m. tìte Cov(X ,Y ) = 0

Sundiaspor�

Cov(X ,Y ) = E ((X − µX )(Y − µY )) = E (XY )− µXµY

Sundiaspor�

Cov(X ,Y ) = E ((X − µX )(Y − µY )) = E (XY )− µXµY

Idiìthtec diaspor�c

An α kai β pragmatikoÐ arijmoÐ tìte

Var(αX + β) = α2Var(X )

An α kai β pragmatikoÐ arijmoÐ kai X kai Y apì koinoÔ

katanemhmènec t.m. tìte

Var(αX + βY ) = α2Var(X ) + β2Var(Y ) + 2αβCov(X ,Y )

an epiplèon oi t.m. X kai Y eÐnai anex�rthtec tìte

Var(αX + βY ) = α2Var(X ) + β2Var(Y )

Suntelest c susqètishc

ρ =Cov(X ,Y )

σXσY

metr�ei thn grammik ex�rthsh dÔo metablht¸n

ρ kajarìc arijmìc

−1 ≤ ρ ≤ 1

ρ = 1 tèleia jetik susqètish

ρ = −1 tèleia arnhtik susqètish

ρ = 0 asusqètista

An oi t.m. X kai Y eÐnai anex�rthtec tìte ρ = 0To antÐstrofo isqÔei ???

ρ =Cov(X ,Y )

σXσY

−1 ≤ ρ ≤ 1

ρ = 0 asusqètista

An oi t.m. X kai Y eÐnai anex�rthtec tìte ρ = 0

To antÐstrofo isqÔei ???

ρ =Cov(X ,Y )

σXσY

−1 ≤ ρ ≤ 1

ρ = 0 asusqètista

An oi t.m. X kai Y eÐnai anex�rthtec tìte ρ = 0To antÐstrofo isqÔei ???

'Askhsh

X , Y 0 1 2 3 4 fX0 0.405 0.310 0.030 0.012 0.005 0.762

1 0.112 0.030 0.015 0.008 0.004 0.169

2 0.020 0.012 0.009 0.007 0.003 0.051

3 0.006 0.004 0.004 0.003 0.001 0.018

fY 0.543 0.356 0.058 0.030 0.013

UpologÐste thn E(X )

UpologÐste thn Var(X )

UpologÐste thn E(XY )

UpologÐste ρ

'Askhsh

X , Y 0 1 2 3 4 fX0 0.405 0.310 0.030 0.012 0.005 0.762

1 0.112 0.030 0.015 0.008 0.004 0.169

2 0.020 0.012 0.009 0.007 0.003 0.051

3 0.006 0.004 0.004 0.003 0.001 0.018

fY 0.543 0.356 0.058 0.030 0.013

UpologÐste ρ

'Askhsh

X , Y 0 1 2 3 4 fX0 0.405 0.310 0.030 0.012 0.005 0.762

1 0.112 0.030 0.015 0.008 0.004 0.169

2 0.020 0.012 0.009 0.007 0.003 0.051

3 0.006 0.004 0.004 0.003 0.001 0.018

fY 0.543 0.356 0.058 0.030 0.013

UpologÐste ρ

'Askhsh

X , Y 0 1 2 3 4 fX0 0.405 0.310 0.030 0.012 0.005 0.762

1 0.112 0.030 0.015 0.008 0.004 0.169

2 0.020 0.012 0.009 0.007 0.003 0.051

3 0.006 0.004 0.004 0.003 0.001 0.018

fY 0.543 0.356 0.058 0.030 0.013

UpologÐste ρ

'Askhsh

'Estw Q kai Y oi bajmoÐ sugkèntrwshc dÔo rÔpwn kai èstw

f (x , y) =x + y

1000, 0 < x < 10, 0 < y < 10

UpologÐste ρ (ρ = −0.091) Ti shmaÐnei autì???

'Askhsh

f (x , y) =x + y

1000, 0 < x < 10, 0 < y < 10

'Askhsh

f (x , y) =x + y

1000, 0 < x < 10, 0 < y < 10

'Askhsh

f (x , y) =x + y

1000, 0 < x < 10, 0 < y < 10

UpologÐste ρ

(ρ = −0.091) Ti shmaÐnei autì???

'Askhsh

f (x , y) =x + y

1000, 0 < x < 10, 0 < y < 10

Di�mesoc

'Estw Q t.m. me a. sunar. k. F (x)Di�mesoc thc Q lègetai k�je arijmìc δ tètoioc ¸ste

F (δ) = P(X ≤ δ) ≥ 1

2≥ P(X < δ) = F (δ−)

H di�meshc up�rqei p�nta se antÐjesh me th mèsh tim pou

mporeÐ na mhn up�rqei se k�poiec katanomèc.

Di�mesoc

'Estw Q t.m. me a. sunar. k. F (x)Di�mesoc thc Q lègetai k�je arijmìc δ tètoioc ¸ste

F (δ) = P(X ≤ δ) ≥ 1

2≥ P(X < δ) = F (δ−)

H di�meshc up�rqei p�nta se antÐjesh me th mèsh tim pou

mporeÐ na mhn up�rqei se k�poiec katanomèc.

'Otan h Q eÐnai suneq c t.m. me sunar. p.p. f (x), koruf thc Q

lègetai k�je shmeÐo x sto opoÐo h f (x) èqei topikì mègisto.

'Otan h Q eÐnai diakrit t.m. me sunar. p. f (x) kai dunatèctimèc x1 < x2 < . . . < xn < xn+1, mÐa endi�mesh dunat tim xklègetai koruf thc Q ìtan h f (xk) eÐnai megalÔterh apì tic

f (xk−1), f (xk+1), en¸ mÐa akraÐa dunat tim thc Q lègetai

koruf ìtan h tim thc sun. p. se autì to shmeÐo eÐnai

megalÔterh apì th sun. p. sth geitonik dunat tim thc Q.

'Otan h Q eÐnai suneq c t.m. me sunar. p.p. f (x), koruf thc Q

lègetai k�je shmeÐo x sto opoÐo h f (x) èqei topikì mègisto.

'Otan h Q eÐnai diakrit t.m. me sunar. p. f (x) kai dunatèctimèc x1 < x2 < . . . < xn < xn+1, mÐa endi�mesh dunat tim xklègetai koruf thc Q ìtan h f (xk) eÐnai megalÔterh apì tic

f (xk−1), f (xk+1), en¸ mÐa akraÐa dunat tim thc Q lègetai

koruf ìtan h tim thc sun. p. se autì to shmeÐo eÐnai

megalÔterh apì th sun. p. sth geitonik dunat tim thc Q.

PosostiaÐa shmeÐa

'Estw Q t.m. me ajr. sunar. k. F (x) kai èstw p ènac arijmìc

tètoioc ¸ste 0 < p < 1. Opoiod pote shmeÐo xp tètoio ¸ste

P(X < xp] ≤ p ≤ P(X ≤ xp)

isodÔnama

F (x−p ) ≤ p ≤ F (xp)

lègetai p�posostiaÐo shmeÐo thc Q.

'Otan h Q eÐnai suneq c t.m. p�posostiaÐo shmeÐo eÐnai

k�je shmeÐo xp pou ikanopoieÐ thn exÐswsh F (xp) = p

To shmeÐo x0.5 eÐnai h di�mesoc thc Q

Ta shmeÐa x0.25 kai x0.75 eÐnai gnwst� wc pr¸to kai trÐto

tetarthmìrio.

PosostiaÐa shmeÐa

'Estw Q t.m. me ajr. sunar. k. F (x) kai èstw p ènac arijmìc

tètoioc ¸ste 0 < p < 1. Opoiod pote shmeÐo xp tètoio ¸ste

P(X < xp] ≤ p ≤ P(X ≤ xp)

isodÔnama

F (x−p ) ≤ p ≤ F (xp)

lègetai p�posostiaÐo shmeÐo thc Q.

'Otan h Q eÐnai suneq c t.m. p�posostiaÐo shmeÐo eÐnai

k�je shmeÐo xp pou ikanopoieÐ thn exÐswsh F (xp) = p

To shmeÐo x0.5 eÐnai h di�mesoc thc Q

Ta shmeÐa x0.25 kai x0.75 eÐnai gnwst� wc pr¸to kai trÐto

tetarthmìrio.

pijan'othtes kai statistik'h december 21, 2010 · mèsh tim tuqaÐwn metablht¸n e(x) =...

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