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1 Pertemuan 05 Sebaran Peubah Acak Diskrit Mata kuliah: A0392 - Statistik Ekonomi Tahun: 2010.

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Presentasi berjudul: "1 Pertemuan 05 Sebaran Peubah Acak Diskrit Mata kuliah: A0392 - Statistik Ekonomi Tahun: 2010."— Transcript presentasi:

1 1 Pertemuan 05 Sebaran Peubah Acak Diskrit Mata kuliah: A Statistik Ekonomi Tahun: 2010

2 2 Outline Materi: Ruang Sampel, Konsep dasar peluang dan Peubah Acak Nilai harapan peubah acak Peluang Bersyarat dan Bebas Peluang Total dan Kaidah Bayes

3 3 Konsep Dasar Peluang Basic Probability Concepts –Sample spaces and events, simple probability, joint probability Conditional Probability –Statistical independence, marginal probability Bayes’ Theorem Counting Rules

4 4 Sample Spaces Collection of All Possible Outcomes –E.g., All 6 faces of a die: –E.g., All 52 cards of a bridge deck:

5 5 Kejadian (Events) Simple Event –Outcome from a sample space with 1 characteristic –E.g., a Red Card from a deck of cards Joint Event –Involves 2 outcomes simultaneously –E.g., an Ace which is also a Red Card from a deck of cards

6 6 Visualizing Events Contingency Tables Tree Diagrams Red Black Total Ace Not Ace Total Full Deck of Cards Red Cards Black Cards Not an Ace Ace Not an Ace

7 7 Contingency Table A Deck of 52 Cards Ace Not an Ace Total Red Black Total Sample Space Red Ace

8 8 Full Deck of Cards Tree Diagram Event Possibilities Red Cards Black Cards Ace Not an Ace Ace Not an Ace

9 9 Probability Probability is the Numerical Measure of the Likelihood that an Event Will Occur Value is between 0 and 1 Sum of the Probabilities of All Mutually Exclusive and Collective Exhaustive Events is 1 Certain Impossible.5 1 0

10 10 (There are 2 ways to get one 6 and the other 4) E.g., P ( ) = 2/36 Computing Probabilities The Probability of an Event E: Each of the Outcomes in the Sample Space is Equally Likely to Occur

11 11 Computing Joint Probability The Probability of a Joint Event, A and B:

12 12 P(A 1 and B 2 )P(A 1 ) Total Event Joint Probability Using Contingency Table P(A 2 and B 1 ) P(A 1 and B 1 ) Event Total 1 Joint Probability Marginal (Simple) Probability A1A1 A2A2 B1B1 B2B2 P(B 1 ) P(B 2 ) P(A 2 and B 2 ) P(A 2 )

13 13 Computing Compound Probability Probability of a Compound Event, A or B:

14 14 P(A 1 ) P(B 2 ) P(A 1 and B 1 ) Compound Probability (Addition Rule) P(A 1 or B 1 ) = P(A 1 ) + P(B 1 ) - P(A 1 and B 1 ) P(A 1 and B 2 ) Total Event P(A 2 and B 1 ) Event Total 1 A1A1 A2A2 B1B1 B2B2 P(B 1 ) P(A 2 and B 2 ) P(A 2 ) For Mutually Exclusive Events: P(A or B) = P(A) + P(B)

15 15 Computing Conditional Probability The Probability of Event A Given that Event B Has Occurred:

16 16 Conditional Probability Using Contingency Table Black Color Type Red Total Ace 224 Non-Ace Total Revised Sample Space

17 17 Conditional Probability and Statistical Independence Conditional Probability: Multiplication Rule:

18 18 Conditional Probability and Statistical Independence Events A and B are Independent if Events A and B are Independent When the Probability of One Event, A, is Not Affected by Another Event, B (continued)

19 19 The Law of Total Probability P(A) = P(A  S 1 ) + P(A  S 2 ) + … + P(A  S k ) = P(S 1 )P(A|S 1 ) + P(S 2 )P(A|S 2 ) + … + P(S k )P(A|S k ) P(A) = P(A  S 1 ) + P(A  S 2 ) + … + P(A  S k ) = P(S 1 )P(A|S 1 ) + P(S 2 )P(A|S 2 ) + … + P(S k )P(A|S k ) Let S 1, S 2, S 3,..., S k be mutually exclusive and exhaustive events (that is, one and only one must happen). Then the probability of another event A can be written as

20 20 The Law of Total Probability (Cont.) A A  S k A  S 1 S 2…. S1S1 SkSk P(A) = P(A  S 1 ) + P(A  S 2 ) + … + P(A  S k ) = P(S 1 )P(A|S 1 ) + P(S 2 )P(A|S 2 ) + … + P(S k )P(A|S k ) = Σ n i=1 P(S i )P(A|S i ) P(A) = P(A  S 1 ) + P(A  S 2 ) + … + P(A  S k ) = P(S 1 )P(A|S 1 ) + P(S 2 )P(A|S 2 ) + … + P(S k )P(A|S k ) = Σ n i=1 P(S i )P(A|S i )

21 21 Bayes ’ Rule (Bayes’ Theorem) Let S1, S2, S3,..., Sk be mutually exclusive and exhaustive events with prior probabilities P(S1), P(S2),…,P(Sk). If an event A occurs, the posterior probability of Si, given that A occurred is

22 22 We know: P(F) = P(M) = P(H|F) = P(H|M) = We know: P(F) = P(M) = P(H|F) = P(H|M) = Example From a previous example, we know that 49% of the population are female. Of the female patients, 8% are high risk for heart attack, while 12% of the male patients are high risk. A single person is selected at random and found to be high risk. What is the probability that it is a male? Define H: high risk F: female M: male

23 23 Bayes’ Theorem Using Contingency Table 50% of borrowers repaid their loans. Out of those who repaid, 40% had a college degree. 10% of those who defaulted had a college degree. What is the probability that a randomly selected borrower who has a college degree will repay the loan?

24 24 Bayes’ Theorem Using Contingency Table (continued) Repay College Total

25 Review probabilitas (Tambahan Materi)

26 26 Sample space, sample points, events Sample space, , adalah sekumpulan semua sample points,  yang mungkin; dimana  –Contoh 1. Melemparkan satu buah koin:  ={Gambar,Angka} –Contoh 2. Menggelindingkan dadu:  ={1,2,3,4,5,6} –Contoh 3. Jumlah pelanggan dalam antrian:  ={0,1,2,…} –Contoh 4. Waktu pendudukan panggilan (call holding time):  ={x  x>0} Events A,B,C,…   adalah himpunan bagian dari sample space –Contoh 1. Angka genap pada sebuah dadu:A={2,4,6} –Contoh 2. Tidak ada pelanggan yang mengantri : A={0} –Contoh 3. Call holding time lebih dari 3 menit. A={x  x>3} Event yang pasti : sample space  Event yang tidak mungkin : himpunan kosong (  )

27 27 Kombinasi event Union (gabungan) :“A atau B” : A  B={  A atau  B} Irisan: “A dan B” : A  B={  A dan  B} Komplemen : “bukan A”:A c ={  A} Event A dan B disebut tidak beririsan (disjoint) bila : A  B=  Sekumpulan event {B1,B2,…} merupakan partisi dari event A jika –(i) B i  B j =  untuk semua i  j –(ii)  i B i =A

28 28 Probabilitas (peluang) Probabilitas suatu event dinyatakan oleh P(A) P(A)  [0,1] Sifat-sifat peluang Back to Six

29 29 Conditional Probability (Peluang bersyarat) Asumsikan bahwa P(B)>0 Definisi : Conditional probability dari suatu event A bila diketahui event B terjadi didefinisikan sebagai berikut Dengan demikian

30 30 Teorema Probabilitas Total Bila {B i } merupakan partisi dari sample space  Lalu {A  B i } merupakan partisi dari event A, maka berdasarkan sifat probabilitas yang ketujuh pada slide nomor 28 Kemudian asumsikan bahwa P(B i )>0 untuk semua i. Maka berdasarkan uraian pada slide nomor 29 dapat didefinisikan teorema probabilitas total sbbnomor 29

31 31 Teorema Bayes Bila {B i } merupakan partisi dari sample space  Asumsikan bahwa P(A)>0 dan P(B i )>0 untuk semua i. Maka berdasarkan uraian pada slide nomor 29 Kemudian, berdasarkan teorema probabilitas total, kita peroleh Ini merupakan teorema Bayes –Peluang P(B i ) disebut peluang a priori dari event B i –Peluang P(B i  A) disebut peluang a posteriori dari event B i (bila diketahui event A terjadi)

32 32 Kesalingbebasan statistik dari event (Statistical independence of event) Definisi : Event A dan B saling bebas (independent) jika Dengan demikian Demikian pula

33 33 Peubah acak (random variables) Definisi : Peubah acak X (yang merupakan bilangan riil [real-valued]) adalah fungsi bernilai riil dan dapat diukur yang didefinisikan pada sample space  ;X:    –Setiap titik sample (sample points)  dihubungkan dengan sebuah bilangan riil X(  ) –Dengan kata lain : memetakan setiap titik sample ke sebuah bilangan riil menggunakan peubah acak X

34 34 Sebuah koin dilempar tiga kali; setiap lemparan akan menghasilkan head (H) atau tail (T) Sample space: Misalnya peubah acak X merupakan jumlah total tail (T) dalam ketiga eksperimen pelemparan koin tersebut, maka : Contoh


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