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*Orismos ths pi8anothtas

kata Laplace.



1. Gia ka8e endexomeno

 0P(A)1

    _

2.P(A)=1-P(A)

3.P(AUB)=P(A)+P(B)-P(A.B)



*Apodei3h a8roistikou typou.



 Exoume AUB=A.B'UB 

(me (A.B').B= (dhladh asymbibasta)



 Epishs exoume

P(A.B'UB)= P(A.B')+ P(B)= P(AUB).



 Epishs P(A.B')= P(A)- P(A.B)



 Opote P(AUB)= P(A)+P(B)-P(A.B)



*Tautothtes de morgan.

  _ _  ___

1.AUB= A.B

  _ _  ___

2.A.B= AUB



*Asymbibasta endexomena.



A,B }

A.B= }=> P(AUB)=P(A)+P(B)



*Desmeumenh pi8anothta.



Desmeumenh pi8anothta tou A

dedomenou oti exei pragmatopoih8ei

to B.



        P(A.B)

P(A/B)=--------

         P(B)



Idiothtes:



1. P(A/B)=0 ==>A.B=

     _

2. P(A/B)=1-P(A/B)

       _

   P(A/B)1-P(A/B)



3. P[ (AUB)/C ]=

 =P(A/C)+P(B/C)-P[(A.B)/C]



4.An ta endexomena A,B einai 

symbibasta kai to 

_

B den einai adynaton:



    _    P(A)

P(A/B)=--------

        1-P(B)



5.

*Ane3arthta endexomena.



Oi parakatw treis sxeseis einai

isodynames:



a) P(A)=P(A/B)

b) P(B)=P(B/A)

c) P(A.B)=P(A).P(B).



Asymbibasta KAI ane3arthta

gegonota:



Ave3arthta:  

P(A.B)=P(A).P(B)}

Asymbibasta:    >==>

A.B=           }



==>P(A).P(B)=0 ==> 

==>A= h B=



*Ane3arthsia N endexomenwn.



Statistikh plhrhs ane3arthsia.



1. P(Ak.A)=P(Ak).P(A)

Gia olous k dynatous

syndiasmous.



2. P(Ak.A.A)=P(Ak).P(A).P(A)

Gia olous k dynatous

syndiasmous.



 ................................



v-1. P(A1.A2....Av)=

=P(A1).P(A2)...P(Av)



*Pollaplasiastikos typos



O pollaplasiastikos typos 

ypologizei thn pi8anothta ths 

tomhs v endexomenwn, ta opoia 

prepei na katatassontai kata

xronikh seira.



P(A1.A2...Av)=

=P(A1).P(A2/A1).P(A3/A1.A2)....P(Av/A1.A2...Av-1)



*Typos olikhs pi8anothtas

-Typos Bayes.



O typos ths olikhs pi8anothtas 

briskei th pi8anothta tou

endexomenou pou einai deutero 

sth xronikh e3eli3h

(adesmeuth pi8anothta).



  Enw o typos tou Bayes briskei 

th pi8anothta pou einai prwth 

sth xronikh e3eli3h

(desmeumenh pi8anothta).



8ewroume ta B1,B2,...Bv ana dyo 

asymbibasta 

Bk.B=  V k,=1,2,..v.

Kai to endexomeno A to opoio 

symbainei pantote se syndiasmo 

me ena ap'ta 

B1,B2...Bv tote:



1.Typos olikhs pi8anothtas

P(A)=P(B1).P(A/B1)+P(B2).P(A/B2)+...+P(Bv).P(A/Bv)



2.Typos Bayes

                         P(Bk).P(A/Bk)

P(Bk/A)=---------------------------------------------

        P(B1).P(A/B1)+P(B2).P(A/B2)+...+P(Bv).P(A/Bv)





*Mh epanalhptikoi syndiasmoi

twn v stoixeiwn ana k.



To plh8os twn syndiasmwn pou 

dhmiourgoume otan apo ena synolo

v stoixeiwn kanoume e3agwges 

k stoixeiwn xwris epanatopo8ethsh

einai:



 (v)     v!

 (k)=----------

      k!.(v-k)!



*Synarthsh katanomhs:



F(t)=P(xt)=   P(x=xk)

            xkt



             t

F(t)=P(xt)= f(x).dx ==>

            a



==> f(x)= F'(x)

Idiothtes:



1. 0F(t)1

2. F(t)=0 gia t<xmin

3. F(t)=1 gia t>xmax

4. lim F(t)=1, lim F(t)=0

   t         t-



3. P(a<x)=F()-F(a).



*Synarthsh pi8anothtas.



Gia na einai h f(x) synarthsh 

pi8anothtas prepei:



1. 0f(x)1  V x=1,2,...



   

2.   f(x)=1.

   x=1



(Sta x=1,2, kai 

               x=1

to "1" einai sxetiko dhladh einai

to xmin sthn pragmatikothta).



*f(x)=1/(x.(x+1))



       1       1     1

f(x)=-------= --- - ---

     x.(x+1)   x    x+1



*Synarthsh Pyknothtas Pi8anothtas



Idiothtes:



1. f(x)0  V x  R



     

2.   f(x).dr=1

    a



3. f(x)=F'(x)

             

4. P(<x<)= f(x).dx=F()-F().

            



*x:Diamesos tyxaias metablhths.



P(xx)= 0.5= P(x>x)



      h

 x

  f(x).dx=0.5

a

      h



f(x)=



*xp:Posostiaio shmeio.



P=P(xxp).



      h



F(xp)=P



*x:Koryfh.



To shmeio sto opoio h synarthsh 

pi8anothtas parousiazei megisto 

to opoio einai kai monadiko.



*Apo fx(x) se fy(y).



 Estw h tyxaia metablhth me 

synarthsh pyknothtas

fx(x).

 Estw h tyxaia metablhth

y=(x),opoy (x) synexhs

synarthsh.

 Zhtame na prosdiorisoume th 

synarthsh pi8anothtas 

ths y dhladh fy(y).



1. Lynw ws pros x thn y=f(x)

kai estw x1=x1(y),

x2=x2(y) oi lyseis.



2.Briskoume thn paragwgo

ths (x)==> y'='(x).



3.Antika8istoume:

        fx(x1(y))     fx(x2(y))           fx(xv(y))

fy(y)= ----------- + ----------- + ... + -----------

       |'(x1(y))|   |'(x2(y))|         |'(xv(y))|



4.Briskoume tis times

pou pairnei h y basei

twn timwn tou x.



*Kanonikh katanomh.



H metablhth x akolou8ei

kanonikh katanomh sto

(,) otan:

       1

f(x)=----- , <x<

      -



*Ek8etikh katanomh.



H metablhth x akolou8ei

ek8etikh katanomh me

parametro  otan:



        -x

f(x)=.e    , x>0



*Mesh timh .



1.Tyxaia metablhth apari8mhth:



=E(x)



E(x)=x1.P1+x2.P2+...xv.Pv=



  v

=  xi.Pi

 i=1



E(h(x))=h(x1).P1+h(x2).P2+...+h(xv).Pv





2.Tyxaia metablhth synexhs:



=E(x)



      

E(x)= x.f(x).dx

     

         

E(h(x))= h(x).f(x).dx

        



*Idiothtes meshs timhs.



1.E(x+c)=E(x)+c

2.E(c.x)=c.E(x)

3.E(x+y)=E(x)+E(y).





*:Topikh apoklish.

Einai h 8etikh

tetragwnikh riza

ths diasporas.



=Vav(x)



*r:Roph ta3hs r.



        r  + r

   =E(x )= x.f(x) dx

   r      -



*Kentrikh roph ta3hs v



             v

   =E{ (X-)  }

   v



*Diaspora tyxaias metablhths.



=V(X) =E{ (X-) }=

=E{ [X-E(x)] }=

=E(x)-E(x)



Idiothtes:



Idiothtes:



1. An X=c , c  R

tote V(X)=0



2. An ,  R tote

V(.X+)=.V(X)



3. V(X+Y)V(X)+V(Y)



4. V(X)=E(X)-E(X)





*Synarthsh Gamma (G).



     + p-1  -x

G(p)= x   .e  .dx

     0



Idiothtes:



1. G(p+1)=p.G(p)

2. G(n+1)=n!, n  N.



*Anisothta Chebyshev



a) (x)=(-x) xR

b) (x)>0 x(0,+)

c)  au3ousa sto (0,+)

d E{(X)} peperasmenh



            E{(x)}

P(|X|x)  --------- ,

              (x)



 x>0



*Syndiakymansh Cov



Cov(X,Y)=E{(X-x).(Y-y)}



Idiothtes:

1. Cov(X,C)=0 , C t.m ish

  me c



2. Cov(X,Y)=Cov(Y,X)



3. Cov(X+Z,Y)=

  =Cov(X,Z)+Cov(Y+Z)



4. Cov(.X+,.Y+)=

  =..Cov(X,Y) ,

   ,  R - sta8era.



5. Cov(X,X)=V(X).

 

6. Cov(X,Y)=

 =E(X.Y)-E(X).E(Y)



*Anisothta Cauchy-Swarz.



 E(|X.Y|)

                 

 [ E(X).E(Y) ]



* Syntelesths sysxetishs



           Cov (X,Y)

 (X,Y) = ------------

           V(X).V(Y)



 , -11



Idiothtes:



1.(X,X)=1

2. ,  R, .0 =>



  (.X+,.Y+)=

 (X,Y).e,



    { 1 an .>0

  e={

    { -1 an .<0



*Diwnymikh katanomh

 Estw ena peirama ekteleitai n 

fores me tis diadoxikes dokimes 

na einai meta3y tous ane3arthtes.

 x:ari8mos twn epityxiwn stis 

   v ane3arthtes dokimes tote:



        (v)  k  v-k

P(x=k) =(k).p .q    ,k=0,1,2,..v



p:pi8anothta epityxias.

q:pi8anothta apotyxias.

 p+q=1



*Gewmetrikh katanomh.

 Estw akolou8ia ane3arthtwn 

dokimwn enos peiramatos pou

termatizei otan emfanistei gia 

prwth fora epityxia.

 x:o ari8mos twn apaitoumenwn

   dokimwn.



            k-1

  P(x=k)=p.q   ,k=1,2,3,...v



*Katanomh Pascal.

 Estw akolou8ia ane3arthtwn 

dokimwn pou termatizei otan

emfanistei kai h v-osti epityxia.

 x:ari8mos twn apaitoumenwn 

   dokimwn.



        (k-1)  v  k-v

 P(x=k)=(v-1).p .q



   k=v,v+1,v+2,...



4.Ypergewmetrikh katanomh.

 Estw ena kouti pou periexei 

 aspres kai  maures sfaires.

 Bgazoume v sfaires.

 x:einai oi aspres.



 Sthn periptwsh me

epanatopo8ethsh oi

dokimes einai ane3arthtes

ara h katanomh tou x

8a einai downymikh.



        (v)       k       v-k

 P(x=k)=(k).(-----) .(-----)

              +      +



 k=0,1,2,....v



 Xwris epanatopo8ethsh:

         ()   (  )

         (k) . (v-k)

 P(x=k)=--------------

           (+)

           ( v )



  max{0,v-}kmin{v,}.



*Katanomh Poison.



                k

          -   

  P(x=k)=e  .-----

               k!



  k=0,1,2.......



>0:Mesos oros gegonotwn

    ana monada.



Mesh timh katanomhs Poison:



                   k

           -    

E(X)=   k.e   . ---- =

     k=0          k!



               k-1

   -           

= e  . .   -------= 

         k=1  (k-1)!



  -     

=e  .. e =







                              k

                     -     

E(X(X-1))=   k.(k-1).e   . ---- =

           k=2               k!



               k-2

   -   2        

= e  . .    -------= 

          k=2  (k-2)!



  -  2    2

=e  .. e =





Diaspora Poisson

                       2

E(X^2)=E(X(X-1))+E(X)= + 



                     2       2

V(X)=E(X^2)-E(X)^2=  +  -  = 





EMP poisson



              xi            (1,n)xi

      n  -         -n   

L()=  e   ---   = e    ----

     i=1     xi!         (1,n)xi!





l()= -n + ((1,n)xi).ln - (1,n)ln(xi!)



d             1

-- l()= -n + - (1,n)xi = 0

d            



    1

=  - (1,n)xi = M(x)

    n





*Kanonikh katanomh.



          1       -(x-)/(2)

  f(x)=--------.e

        .(2)



    -<x<+



 :mesh timh ths x

 :typikh apoklish ths x



 Idiothtes:



1.

x1~N(m1,1).

x2~N(m2,2).

x1,x2 ane3arthtes

x2-x1~N(m2-m1,(1+2)



*Typopoihmenh kanonikh

 katanomh:

  Gia =0, =1:



               1     -x

  f(x)=(x)=-------.e

             (2)



  Idiothtes:

 Estw h tyxaia metablhth

x akolou8ei kanonikh

katanomh me parametrous

( kai ).

 Tote h y=(x-)/

akolou8ei typopoihmenh

katanomh me parametrous

(0,1).



     x-

  y=-----

      

          x-

  F(x)=(-----)

           



  (-c)= 1-(c).



*Diasthma empistosynhs



 Exoume:

 S1,S2 amerolhptes 

deigmatikes diaspores.



 Tote h katanomh ths



 (S1/1)/(S2/2)

 einai h F(n -1,n -2).

            1    2



 Diasthma empistosynhs

tou logou 1/2.



*Krithrio X(Elegxos ane3arthsias)



0ij: Oi syxnothtes pou mas 

dinontai.



Eij: Anamenouses syxnothtes



 |             | Synolo

-+-------------+--

 | n11 n12 n13 | n

 |             | .

 | n21 n22 n23 | ni

 |             | .

 |             | nr

-+-----------------

S| n1... nj..ns| N





      ni.nj 

Eij= -------

        N



Otan E  < 5 tote sympthsoume

      ij



thn orizontia grammh i me thn i+1

(a8roizontas ta antistoixa ka8eta

stoixeia), kai 3anapypolo ta 

alagmena Eij gia ton kainourgio

pinaka. (A8roizoume ta 

E   + E      ). Ta ypoloipa

 ij    (i+1)j



Eij den allazoune.





       (0ij - Eij)

X=  --------------

   ij      Eij



Ba8moi eleu8erias:



 v=(r-1).(s-1)



(Blepe nr,ns ston poio panw 

pinaka).



Ypo8esh H0: Ta dedomena tou

orizontiou a3ona ston pinaka

DEN sxetizontai me auta tou 

ka8etou otan:



 X(ypologizomeno) < X

                      v,a



Opou a: epipedo shmantikothtas.

 (P=1-a ston pinaka 3 tou 

bibliou). 

 v: Ba8moi eleu8erias.



*Krithrio X(Elegxos omoiogenias)



Idia me ta poio panw ektos:



Ypo8esh H0: Oi ka8etes sthles

tou pinaka sysxetizontai

meta3y tous.



 H ypo8esh H0 aporiptetai otan



 X(ypologizomeno) > X

                      v,a



Parathrhsh: Otan kapoia ap'tis

steiles einai synarthsh katanomhs

, tote ypologizoume thn synarthsh

katanomhs gia ta diasthmata pou

dinontai kai thn pollaplasiazoume

me to synolo twn syxnothtwn (

kanonikopoihsh sto diasthma pou 

exoume). Kai auth einai h Eij!!



* Grammikh Palindromhsh



Cxx = (x-M(x))^2



Cyy = (y-M(y))^2



Cxy = (x-M(x)).(y-M(y))



r = Cxy/ sqroot(Cxx.Cyy)



s^2 y|x = (Cyy/n-2).(1-r^2)



a = M(y)-b.M(x)



b = Cxy/Cxx 



Genika s^2 = 1/(n-1) Sigma(1,n),(xi-M(x))^2



Syntelesths metablhtothtas L=100.s/M(x) %







