sebaran peubah acak
TRANSCRIPT
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PELUANG &
PEUBAH ACAK
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Ruang Contoh (Sample Space)
Definisi
Himpunan semua kemungkinan hasil suatu percobaan (S)
Contoh
RC Pelemparan mata uang : S = {A , G}
RC Pelemparan dadu : S = {1,2,3,4,5,6}
RC Berat badan :
S = {Berat,Sedang,Ringan}
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Diagram PohonMata Uang dilempar 3 kaliLemparan I Lemparan II lemparan III Titik Contoh
A
G
AA
G
A
G
A
G
A
G
G
G
A
AAA
AAG
AGA
AGG
GAA
GAG
GGA
GGG
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KejadianHimpunan bagian dari
ruang contoh
Kejadian sederhana Kejadian Majemuk
gabungan beberapakejadian sederhana
hanya terdiri dari satu titik contoh
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Mencacah Titik Contoh
• Kaidah penggandaan
n1 n2 n3 … nk
• Permutasi
• Kombinasi
!)(
!
rn
nPrn
suatu susunan yang dibentuk oleh semua atau sebagian dari sekumpulan benda
pengambilan r benda dari n benda tanpa memperhatikan urutan
!)(!
!
rnr
n
r
nC n
r
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Peluang suatu kejadian
Suatu percobaan mempunyai N hasil percobaan yang berbeda & masing-masing memiliki kemungkinan sama untuk muncul.
Jika tepat n di antara hasil tersebut menyusun kejadian A, maka peluang kejadian A adalah
N
nAP )(
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Aksioma Peluang
• P(S) = 1• P(Ø) = 0
• 0 ≤ P(A) ≤ 1
• P(A) = 1 – P(A)
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Peubah Acak
Suatu fungsi yang memiliki nilai real dan ditentukan oleh setiap elemen dalam
ruang contoh
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Contoh• Percobaan : 2 bola
diambil secara berurutan dari 1 kantong berisi 4 bola merah dan 3 bola putih.
• Hasil-hasil percobaan yang mungkin merupakan elemen Ruang Contoh
• Peubah acak Y menyatakan banyaknya merah yang terambil
• Peubah acak X menyatakan banyaknya putih yang terambil
Ruang Contoh
y x
MM 2 0
MP 1 1
PM 1 1
PP 0 2
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Ruang Contoh
RC. Diskret
Jumlah titik contoh berhingga Nilai dinyatakan dalam bilangan cacah
RC. Kontinyu
Banyak titik contoh tak berhingga Nilai dinyatakan dalam bilangan riil
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Sebaran peluang
Misal, peluang suatu peubah acak dinyatakan sebagai fungsi nilai-nilai x yaitu :
P(X = x) = f(x)
maka himpunan semua pasangan berurutan
[x , f(x)]
disebut fungsi peluang atau sebaran peluang bagi peubah acak X.
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Sebaran Peluang Diskret• Suatu table atau rumus yang mencantumkan semua
kemungkinan nilai suatu peubah acak diskret dan peluangnya
• Contoh :Menentukan sebaran peluang banyaknya sisi gambar bila sebuah uang logam dilemparkan 4 kali Ruang contoh mengandung 24 = 16 titik contoh x sisi gambar dan 4-x sisi angka dapat terjadi dalam
cara, dimana x = 0, 1, 2, 3, 4
Fungsi peluang
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Histogram peluang
0 1 2 3 4
1/16
4/16
6/16
x
f (x)
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SEBARAN PELUANG KONTINU
• Peubah acak kontinu berpeluang nol untuk mengambil tepat salah satu nilainya
• sebaran peluang tidak dapat diberikan dalam bentuk tabel
• peluang dinyatakan dari berbagai selang peubah acak kontinu, misal : P( a < X < b)
• Fungsi peluang digambarkan oleh kurva biasanya disebut fungsi kepekatan peluang.
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Beberapa bentukfungsi kepekatan peluang
x
f(x)
x
f(x)
(a) (b)
(c) (d)
x
f(x)
x
f(x)
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Contoh
Peubah acak kontinu X yang memiliki nilai antara x = 2 dan x = 4 mempunyai fungsi kepekatan peluang
a. Buktikan bahwa P(2 < X < 4) = 1
b. Tentukan P(X < 3,5)
c. Tentukan P(2,4 < X < 3,5)
8
1
xxf )(
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Kurva Sebaran Peluang PA X
f(x) 5/8 4/8 3/8
2 4 x
2
2422)()()(
)(
ff
alasxsejajarsisijumlahluas
12
28
5
8
3
42
)(
)( XP
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7002
518
54
8
3
53
,
),(,
),(
XP
Peluang (X < 3.5)
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NILAI TENGAH PEUBAH ACAK
x x1 x2 … xn
P(X=x) f(x1) f(x2) … f(xn)
• misalkan X adalah peubah acak diskret dengan sebaran peluang pada tabel di atas
• Maka nilai tengah atau nilai harapan bagi X adalah
n
iii xfxXE
1
)()(
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Ragam Peubah Acak
n
ii xfxXE
1
222
Suatu populasi yang pengamatannya terdiri atas nilai-nilai peubah acak X, bila percobaan diulang terus-menerus takhingga kali, tidak hanya memiliki nilai
tengah , tetapi juga ragam.
Ragam populasi ini disebut ragam peubah acak X atau ragam sebenarnya
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Fakta/data/praktek
Conjecture/hypothesis/theory/model
Design experiment
Problem formulation
Testing
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Data Metode analisis
• Pembedaan atas skala data – Kwalitatif– Kwantitatif
• Pembedaan atas kepastian perubahan– Probabilistik– Deterministik
• Pembedaan atas waktu kemunculan datanya– Diskrit– Kontinyu
• Tinjauan pengambilan keputusan– Valid– Reliable– Konsisten
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Valid vs Reliable vs Konsisten
reliable
konsisten
Tidak valid
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Data Kwalitatif
• Skala data– Nominal– Ordinal
• Berikan contoh data yang berskala seperti tersebut di atas!
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Data Kwantitatif
• Skala data– Interval– Ratio
• Berikan contoh data yang berskala seperti tersebut di atas!
• Bagaimana dengan– Suhu– Ph– Kadar gula– Berat besi– korelasi
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Perbandingan kwalitatif dan kwantitatif
• Data mana yang mempunyai bayangan metode analisis lebih mudah? kwantitatif
• Dapatkah data kwalitatif dikwantitatifkan?• Metode apa yang biasa digunakan dalam
menganalisis data:– Kwantitatif?– Kwalitatif?
• Bagaimana jika metode kwantitatif digunakan dalam kasus data kwalitatif?
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Probabilistik vs deterministik
• Kasus dunia nyata (environmental problem) lebih banyak akan bersifat probabilistik, multidimensi, dan kompleks
probabilistik deterministik
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Agriculture Age
Industrial Age
Information Age
Wealth definition
Food Food & Things
Knowledge
People work as
Slaves/serfs
Employees
Partners
People work in Hierarchi
es Bureaucracies
TeamnetsProduction system
One-piece Customization
Mass Production Paradigm
Mass customization Paradigm
Scarcity of resource
Abundance of information
Bio
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1960 1970 1980 1990 2000
I T in
vestm
en
t
year
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know
ledg
e
skill
experience
appl
icat
ion
Education and self development
Training and experimentation
Facilitation and on the job training
Coaching and mentoring
Learning Process
competence
Culture
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Core Capabilities
Implementing and
Integrating
Experimenting
ImportingKnowledge
ProblemSolving
EXTERNAL
PRESENT
INTERNAL
FUTURE
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Metode analisis data deskriptif
• Titik tengah– Mean– Median– Modus– Tream mean
• Penyimpangan– Variance– Range
Interval konfidensi
Skewness?
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Interval (konvensional) vs HPD
Interval (Konvensional) HPD
(Highest Probability Distribution)
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Interval Kepercayaan (1)
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Interval kepercayaan (2)
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HPD (Highest Probability Distribution)Peta Kendali
(1-) x
100%
Batas Kendali Bawah
Batas Kendali Atas
95,0
71,3953
109,4810
97,5
64,4857
110,9149
99,0
55,3356
112,7754
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Analytical Models• Qualitative Approach (for qualitative data)
– Independence Analysis– Proportion Analysis– Analytical Hierarchical Process (AHP)
• Quantitative Approach (for quantitative data)– Forecasting (Smoothing, AR, MA, ARIMA)– Clustering Analysis– Regression Analysis– ANOVA– MRP (Material Requirement Planning)
• Simulation
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MIS Versus DSS
MIS DSS Decision support
providedProvide informationabout the performanceof the organization
Provide informationand decision supporttechniques to analyzespecific problems oropportunities
Information form andfrequency
Periodic, exception,demand, and pushreports and responses
Interactive inquiriesand responses
Information format Prespecified, fixedformat
Ad hoc, flexible, andadaptable format
Informationprocessingmethodology
Information producedby extraction andmanipulation ofbusiness data
Information producedby analytical modelingof business data
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Basic Challenges
Building information systems that can actually fulfill management (decision maker) information requirements.
• Internal versus external data.
• Structured versus unstructured data.
• Anticipated versus unanticipated information.
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Challenges
Integrating DSS and ESS with existing systems:
• Consistency of data
• Availability of data
• Timeliness of data
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Issues• Models validity
– Uncertainty– Time and space scales– Establishing a coherent dialogue between models
• Design of a policy assessment process– Define objectives and constraints– Identify the possible controls– Identify the system reaction to controls– Identify the cost/benefits for control policies– Find efficient solutions
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IT practices
Information Management practices
Information behaviorAnd values
Information orientation :
A comprehensiveof high level ideaof how effectivea company is inusing information
BusinessPerformance
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Types of Decisions
Structured: Repetitive; definite procedure; have a high level of certainty; could even be called routine.
Semi-structured: One or more factors are not structured; the more unstructured, the higher the risk.
Unstructured: Unique; non-routine; has definite uncertainty; requires experience and judgment.
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Unstructured StructuredStructured
• Ad hoc
• Unscheduled
• Summarized
• Infrequent
• Forward looking
• External
• Wide Scope
• Prespecified
• Scheduled
• Detailed
• Frequent
• Historical
• Internal
• Narrow Focus
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Ways to Solve Business Problems
Absolution: Ignore it and hope it will go away.
Dissolution: Redesign to eliminate the problem.
Ressolution: Do something that yields an outcome that is good enough emphasizing past experience.
Solution: Involves research and relies heavily on experimentation, quantitative analysis and both common and uncommon sense.
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Speed
To marketTo market competitive, market competitive, market position, market leadership position, market leadership
To decision To decision consensus, commitment, consensus, commitment, responsive responsive
To task completion productivityTo task completion productivity
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How Are Decisions Made?
Big Deal:
• Research and get as much data as possible.
• Evaluate as long as time permits.
• Consensus decision by key participants.
Little Deal:
• Routine based on past experience (habit).
• Quick decision since consequences are minimal.
• Individual versus consensus decision.
Depends on the significance of the decision.
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Solution Selection Criteria1. Risk: including the odds.
2. Economy of effort: greatest results with least
effort or needed change with least disturbance.
3. Timing: based on urgency which is difficult to
systematize.
4. Limitation of Resources: relative to those
that must carry out the decision. No decision is
better than the people that must carry it out.
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Konsep Variabel OBJEK
• Variabel: suatu peubah yang dibuat untuk dapat diisi dengan suatu nilai
• Nilai sebuah variabel merupakan identitas demensi pandang pada suatu objek
• Objek dapat diterangkan dengan jelas oleh suatu susunan variabel yang lebih banyak
• Atau semakin sedikit variabel dari obej yang diamati semakin tidak jealas objek yang dimaksudkan.
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Univariate vs Multivariate
• Univariate merupakan istilah analisis data statistik yang hanya memandang permasalahan hanya dengan demensi terbatas, yaitu satu demensi.
• Multivariate mengandalkan cara pandang permasalahan statistik dengan multi demensi, dan setiap demensi diduga akan berkorelasi. (jika tiap dimensi tidak berkorelasi, maka akan hanya digolongkan sebagai permasalahan yang multivariabel saja).
• Jadi suatu permasalahan yang dipandang secara multivariabel belum tentu merupakan kasus multivariate, sedangkan kasus multivariate selalu mdipandang secara multivariabel.
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Konsep Derajat bebas
• Derajat bebas (db) memberikan informasi tentang tingkat kebebasan suatu data akan dapat terambil secara random dari kelompoknya
• Semakin besar db maka akan semakin besar kepercayaan bahwa data yang dapat diambil adalah dapat mewakili populasinya
• Semakin besar db semakin besar tingkat kepercayaan dalam pengambilan keputusannya, karena keputusannya didasarkan pada prinsip STRONG LAW LARGE NUMBER THEOREM.
• Semakin besar db berarti semakin bervariasinya suatu sisi pandang demensi analisisnya.