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

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

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

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

Kejadian (Events) Simple Event Joint 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

Visualizing Events Contingency Tables Tree Diagrams Black 2 24 26 Ace Not Ace Total Black 2 24 26 Red 2 24 26 Total 4 48 52 Ace Red Cards Not an Ace Full Deck of Cards Ace Black Cards Not an Ace

A Deck of 52 Cards Contingency Table Red Ace Total Ace Red 2 24 26 Not an Ace Total Ace Red 2 24 26 Black 2 24 26 Total 4 48 52 Sample Space

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

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 1 Certain .5 Impossible

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

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

Joint Probability Using Contingency Table Event Event B1 B2 Total A1 P(A1 and B1) P(A1 and B2) P(A1) A2 P(A2 and B1) P(A2 and B2) P(A2) Total P(B1) P(B2) 1 Marginal (Simple) Probability Joint Probability

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

Compound Probability (Addition Rule) P(A1 or B1 ) = P(A1) + P(B1) - P(A1 and B1) Event Event B1 B2 Total A1 P(A1 and B1) P(A1 and B2) P(A1) A2 P(A2 and B1) P(A2 and B2) P(A2) Total P(B1) P(B2) 1 For Mutually Exclusive Events: P(A or B) = P(A) + P(B)

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

Conditional Probability Using Contingency Table Color Type Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 Revised Sample Space

Conditional Probability and Statistical Independence Multiplication Rule:

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)

The Law of Total Probability Let S1 , S2 , S3 ,..., Sk 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 P(A) = P(A  S1) + P(A  S2) + … + P(A  Sk) = P(S1)P(A|S1) + P(S2)P(A|S2) + … + P(Sk)P(A|Sk)

The Law of Total Probability (Cont.) S2…. S1 Sk A A Sk A  S1 P(A) = P(A  S1) + P(A  S2) + … + P(A  Sk) = P(S1)P(A|S1) + P(S2)P(A|S2) + … + P(Sk)P(A|Sk) = Σni=1 P(Si)P(A|Si)

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

Define H: high risk F: female M: male 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 We know: P(F) = P(M) = P(H|F) = P(H|M) = .49 .51 .08 .12

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?

Bayes’ Theorem Using Contingency Table (continued) Repay Repay Total College .2 .05 .25 .3 .45 .75 College Total .5 .5 1.0

Review probabilitas (Tambahan Materi)

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 ()

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

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

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

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

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

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

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) wW dihubungkan dengan sebuah bilangan riil X(w) Dengan kata lain : memetakan setiap titik sample ke sebuah bilangan riil menggunakan peubah acak X

Contoh 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 :