BOLUM_6

  • View
    694

  • Download
    0

Embed Size (px)

Transcript

David Macpherson

8/12/2005

Blm 2:Olaslk & Istatistik

dmacpher@coss.fsu.eduJump to first page

1. Rakamlar toplam

dmacpher@coss.fsu.eduJump to first page

Title goes here

1

David Macpherson

8/12/2005

Rakamlar toplam = toplam; X aile geliri gibi deikendir. Daha sonra toplam aile geliri N gzlemlerinin deeri

N

i =1

Xi = X 1 + X 2 + ... + XN

dmacpher@coss.fsu.eduJump to first page

Rakamlar toplamSabit zamanlar deikeni toplam is deikenler toplamna eittir.

k i =1 Xi = kX 1 + kX 2 + ... + kXNN

dmacpher@coss.fsu.eduJump to first page

Title goes here

2

David Macpherson

8/12/2005

Rakamlar toplamki deiken gzlemlerinin toplam onlarn dier toplamlarna eittir

N i =1

( Xi + Yi ) =i =1 Xi + i =1 YiN N

dmacpher@coss.fsu.eduJump to first page

Rakamlar toplamN zerindeki sabit gzlemlerin toplam sabit rn ve N e eittir.:

N

i =1

k =kN

dmacpher@coss.fsu.eduJump to first page

Title goes here

3

David Macpherson

8/12/2005

2. Tanmlar

dmacpher@coss.fsu.eduJump to first page

Rastgele DeneyRastgele deney: a process leading to at least two possible outcomes with uncertainty as to which will occur.Sonular zlebilirrnein - lmek, madeni para atmek, desteden kart ekmek sastgele deneydir.

dmacpher@coss.fsu.eduJump to first page

Title goes here

4

David Macpherson

8/12/2005

rnek alanBir deneyin tm olaslk sonularnn kurulumu is rnek ktle olarak tanmlanr. rnekler :madeni para atmak S={H,T}. lm S={1,2,3,4,5,6} 3 sonulu madeni para atmak S={HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}dmacpher@coss.fsu.eduJump to first page

rnek noktalarrnek noktarnek alann sonularnn herbiridir. rnekler :Y veya T 1,2,3,4,5 veya 6 YYY

dmacpher@coss.fsu.eduJump to first page

Title goes here

5

David Macpherson

8/12/2005

SonularAn event is a subset of the sample space.rnekler :Sonu A iki madeni para atldnda yaz gelmesi. Sonu TT sonu A ya aittir.sonu a 3 madeni para atldnda nemli etkiye sahiptir.A={HHH, HHT, HTH, THH}

dmacpher@coss.fsu.eduJump to first page

Ortak zel OlaylarSonular zeldir if thebir olayn sonucu dier olayn sonucunun nne geer.A 2 yazdr ve b 2 baldr.ve bunlar ortaktr. Eer sonu A nne gemezse ve maa as sonu B ise bunlar ortaktr.

dmacpher@coss.fsu.eduJump to first page

Title goes here

6

David Macpherson

8/12/2005

Benzer olaylarn sonularrnek :

Bir olay dier olaya benziyorsa sonular eittir.1/6 olaslkla lme atlmaktr

dmacpher@coss.fsu.eduJump to first page

Ortak ayrntl sonularSonular ortaktr tm olaslklarn sonulardr.Example:HH, HT, TH, TT (2 tura, 2 yaz,1 yaz 1 tura)

dmacpher@coss.fsu.eduJump to first page

Title goes here

7

David Macpherson

8/12/2005

3. Olaslk tanmlar

dmacpher@coss.fsu.eduJump to first page

Klasik tanmlarn olaslk sonularyla deney farzedelim , ve bu sonular are benzer ve ortak sonuludur.rnek ktlede a sonucu kuralm. P(A) Ann olasldr. M sonular olumludur, daha sonraP(A) m/n dr

dmacpher@coss.fsu.eduJump to first page

Title goes here

8

David Macpherson

8/12/2005

Klasik tanm rnei2 madeni parammkn sonular (=n) are TT, TH, HT, HH. 2 madeni paradan A sonucu kuralm.m=1.bundan dolay P(A) = 1/4.

dmacpher@coss.fsu.eduJump to first page

Kalasik rnek tanmlar2 zar atalm, n = 36 mmkn sonular(12,13,14,15,16;21,22,23,24,25,26 ; 31,32,33,34,35,36;41,42,43,44, 45,46; 51,52,53,54,55,56; 61,62,63,64,65,66.)

Farz edelim A sonucu 7 gsterimdedir.m=6 ve bundan dolay P(A) = 6/36 veya 1/6dmacpher@coss.fsu.eduJump to first page

Title goes here

9

David Macpherson

8/12/2005

Amprik tanmlarKlasik tanmda problem - belki deneyin tm sonular eit olmayabilir.Sonularn numaralar sonlu deildir..

Bundan dolay olaslk tanm grecelidir

dmacpher@coss.fsu.eduJump to first page

Greceli frekans rneiBurada ya miktar rneiInches 20 Total Days 10 20 45 20 5 100 Relative Frequency 0.1 0.2 0.45 0.2 0.05 1

nc birletirilmi sklk grublarnn rastgele deeridir

dmacpher@coss.fsu.eduJump to first page

Title goes here

10

David Macpherson

8/12/2005

Amprik olaslk tanmlarSonsuz sklklar olaylarn numaralarnnn basitidir. Greceli sklklar tm olaylar tarafndan blnr. Bu m/n olaslna benzer.Greceli sklklar olaslklar gibi ele alabilirmiyiz , n genise. Bu tanmda, sonular benzer sonulara ve ortak ayrntlara Jump to first page ihtiyac yoktur.

dmacpher@coss.fsu.edu

4. Probability Properties

dmacpher@coss.fsu.eduJump to first page

Title goes here

11

David Macpherson

8/12/2005

Bir sonucun olaslBir sonucun olasl 0 ve 1 arasndadr.0 P(A) 1 Eer P(A)=0 sonu ortaya kmayacaktr. Eer P(A)=1 sonu ortaya kacaktr. Sonucun olasl bu ular arasndadr.dmacpher@coss.fsu.eduJump to first page

Ortak ayrntl sonularEer A, B ve C ortaksa, onlardan birinin olasldier olaslklarn toplamna eit olacaktr.P(A+B+C)= P(A) + P(B) + P(C)

dmacpher@coss.fsu.eduJump to first page

Title goes here

12

David Macpherson

8/12/2005

Ortak zellikler ve ayrntl sonular

Eer A, B, ve C ortak zellikli ve ortak ayrntli sonularsa daha sonra bireysel olaslklarn toplam 1dir.P(A+B+C)=P(A) + P(B) + P(C)= 1

dmacpher@coss.fsu.eduJump to first page

Ortak zellikler ve ayrntl sonular

Throw die ve bundan dolay 1,2,3,4,5,6.drBu sonular ortak zellikli ve ayrntl sonulardir.Birinin olasl: 1/6 Bundan dolay P (1,2 veya 3) = P(1) + P(2) + P(3) = 1/6 + 1/6 + 1/6=1/2 Ve P(1+2+3+4+5+6) = P(1)+ P(2)....=1

dmacpher@coss.fsu.eduJump to first page

Title goes here

13

David Macpherson

8/12/2005

statiksel bamsz sonular

A,B ve C bamsz sonulardr eer onlarn olasl onlarn olaslnn rndr.Eer P(ABC) = P(A)P(B)P(C)P(ABC) birleik olaslk olaral tanmlanr P(A) ve P(B) ve P(C)marjinal artsz ve zel olaslk olarak tanmlanr.

dmacpher@coss.fsu.eduJump to first page

statiksel bamsz sonu rnei

2 zar attnda 6y salamann olasl sallamadrBir alt olasl A Bir alt olasl B Bundan dolay P(AB) = P(A) P(B) = (1/6)(1/6)= 1/36

dmacpher@coss.fsu.eduJump to first page

Title goes here

14

David Macpherson

8/12/2005

artl olaslkfarzedelim sonu B nin olasln istiyoruz bilinen A meydana geldimi?Bu artl olaslktrP(B\A) = P(AB)/P(A)

dmacpher@coss.fsu.eduJump to first page

artl olaslk rneiBir desteden kart seileceini farz edelim.sinek kral kma olasl nedir?Sonu A kral sonu B ise sinek P(B/A) = P(AB)/P(A)P(B/A) = 1/52 / 4/52 = 1/4

dmacpher@coss.fsu.eduJump to first page

Title goes here

15

David Macpherson

8/12/2005

5. Olaslk sklk fonksiyonu

dmacpher@coss.fsu.eduJump to first page

Olaslk dalm fonksiyonu

Olaslk younluk fonksiyonu (PDF) rastgele deikenler olasln belirler. Rastgele deikenler iin X, pdf f(X) anlamna gelir. pdf x deerlerinin karsndak olaslklarn dalmn gsterir. Pdf nin toplam 1 dir.

dmacpher@coss.fsu.eduJump to first page

Title goes here

16

David Macpherson

8/12/2005

Olaslk dalm fonksiyon rnei2 madeni para atalm ve 4 sonu elde ederiz.YY TT TY YT imdi, 2 madeni para atldnda rastgele deikenlerin tanmlayalmX 0 1 2 f(X) .25 .50 .25

dmacpher@coss.fsu.eduJump to first page

Discrete Probability Distribution Function

The discrete PDF of X is f(x) =P(X=xi) for i=1,2,...n and f(x) = 0 for X xi where P(X= xi) is probability that the discrete random variable takes the value of xi. Bu rastgele deikenler byk harfalerle tanmlanmtr X, Y etc. (and dierleri kk harf, x, y, etc.).

dmacpher@coss.fsu.eduJump to first page

Title goes here

17

David Macpherson

8/12/2005

Discrete Probability Distribution Functionf(X) .50 .25

Below is the discrete PDF for the two tails example

.25

0

1

2

X

dmacpher@coss.fsu.eduJump to first page

PDF of a Continuous VariableIn this case,araln stndeki rast gele deikenlerin olasln leriz.Sonularn snrsz numaralarf(X)

.45 .20 .10 .05 19 Inches of Rain .20

dmacpher@coss.fsu.eduJump to first page

Title goes here

18

David Macpherson

8/12/2005

CDF F(X)=P(X x) dir veya rastgele deikenlerin olasl X kk x ten daha byk deer tar.Forml , F(X)=xf(x)

Kmlatif younluk fonksiyonu (CDF)

dmacpher@coss.fsu.eduJump to first page

Multivariate PDFEer sonular bir rastgele deikenden daha fazla yaylmsa, daha sonra farkl olaslk dalm fonksiyonlarna sahip oluruz. Bivariate PDF iki rastgele deiken yaylr. formul, the discrete joint PDF is f (X,Y) = P(X=x ve Y=y) ve f (X,Y) = 0 when X x and Y ydmacpher@coss.fsu.eduJump to first page

Title goes here

19

David Macpherson

8/12/2005

Farkl PDF fonksiyon rnei

X in birleik deeri (kolej eitimi) veY (aile geliri).X = 1eer head kolej eitimi ise ; 0 aksi takdirde Y = $5,000 $10,000 or $15,000Y $5,000 $10,000 $15,000 Total 0 24 12 16 52 X 1 0 12 32 44 Total 24 24 48 96Jump to first page

dmacpher@coss.fsu.edu

farkl PDF rneiGrecelisklklarn dnm (olaslklar).Y $5,000 $10,000 $15,000 Total 0 0.250 0.125 0.167 0.542 X 1 0.000 0.125 0.333 0.458 Total 0.250 0.250 0.500 1.000

This is a discrete bivariate PDF. Each cell is a joint probability of college education and income.dmacpher@coss.fsu.eduJump to first page

Title goes here