Fungsi Distribusi Probabilitas Diskrit

Fungsi Distribusi Probabilitas Diskrit Type of model Representation of some underlying phenomenon Mathematical formula Represents discrete random variable Used to get exact probabilities P ( X = x ) = - l x e l x !

Model Distribusi Probabilitas Diskrit Hiper- Binomial Binomial Poisson Geometrik Negatif

Distribusi Binomial

Model Distribusi Probabilitas Diskrit Hiper- Binomial Binomial Poisson Geometrik Negatif

Distribusi Binomial Banyaknya (#) ‘sukses’ dalam n pengamatan/observasi (trial) # merah dalam 15 putaran roda roulette # barang rusak diantara 5 barang # jawaban benar dalam 33 soal ujian # pelanggan diantara 100 yang masuk ke toko

Sifat Distribusi Binomial Hasil dua metode sampling berbeda Populasi tak hingga, tanpa pengembalian Populasi berhingga, dengan pengembalian Gabungan dari n trial yang sama dan saling independen/bebas Setiap trial mempunyai 2 outcome ‘Sukses’ (outcome yang menjadi perhatian) atau ‘Gagal’ Probabilitas dalam setiap trial sama

Fungsi Distribusi Probabilitas Binomial P(X) = Probabilitas dari X (banyaknya ‘sukses’) n = Ukuran sampel p = Probabilitas ‘sukses’ dalam trial x = Banyaknya ‘sukses’ dalam sampel (X = 0, 1, 2, ..., n)

Contoh Distribusi Probabilitas Binomial Eksperimen: Lempar 1 mata uang 4 kali. Amati # T (tail). Berapakah probabilitas dapat 3 T?

Karakteristik Distribusi Binomial n = 5 p = 0.1 Rerata Distribution has different shapes. 1st Graph: If inspecting 5 items & the probability of a defect is 0.1 (10%), the probability of finding 0 defective item is about 0.6 (60%). If inspecting 5 items & the probability of a defect is 0.1 (10%), the probability of finding 1 defective items is about .35 (35%). 2nd Graph: If inspecting 5 items & the probability of a defect is 0.5 (50%), the probability of finding 1 defective items is about .18 (18%). Note: Could use formula or tables at end of text to get probabilities. Deviasi Standar n = 5 p = 0.5

Soal Distribusi Binomial You’re a telemarketer selling service contracts for Macy’s. You’ve sold 20 in your last 100 calls (p = .20). If you call 12 people tonight, what’s the probability of A. No sales? B. Exactly 2 sales? C. At most 2 sales? D. At least 2 sales? Let’s conclude this section on the binomial with the following Thinking Challenge.

Penyelesaian Distribusi Binomial A. P(0) = .0687 B. P(2) = .2835 C. P(at most 2) = P(0) + P(1) + P(2) = .0687 + .2062 + .2835 = .5584 D. P(at least 2) = P(2) + P(3)...+ P(12) = 1 - [P(0) + P(1)] = 1 - .0687 - .2062 = .7251 From the Binomial Tables: A. P(0) = .0687 B. P(2) = .2835 C. P(at most 2) = P(0) + P(1) + P(2) = .0687+ .2062 + .2835 = .5584 D. P(at least 2) = P(2) + P(3)...+ P(12) = 1 - [P(0) + P(1)] = 1 - .0687 - .2062 = .7251

Distribusi Hipergeometrik

Model Distribusi Probabilitas Diskrit Hiper- Binomial Binomial Poisson Geometrik Negatif

Hipergeometrik Distribusi # ‘sukses’ dalam sampel dengan n observasi (trial) Data sampel diambil dari populasi berhingga tanpa pengembalian Contoh: # barang rusak diantara 5 barang Yang rusak tidak dikembalikan

Distribusi Hipergeometrik A ! (N - A) ! × x ! ( A - x ) ! (n - x) ! (N - A) - (n - x) ! P ( X ) = N ! n ! (N - n) ! P(X) = Probabilitas dapat X ‘sukses’ A = # ‘sukses’ dalam populasi x = # ‘sukses’ dalam sampel n = Ukuran sampel N = Ukuran Populasi

Distribusi Binomial Negatif

Model Distribusi Probabilitas Diskrit Distkrit Hiper- Binomial Binomial Poisson Geometrik Negatif

Distribusi Negatif Binomial # trial until sampai mendapat ‘sukses’ pertama Setiap trial saling independen Setiap trial mempunyai probabilitas ‘sukses’ p Contoh: # banyaknya barang diperiksa (n) sebelum mendapat barang (x) rusak

Fungsi Distribusi Binomial Negatif - 1) ! = x - n - x P ( n ) p ( 1 p ) (x - 1) ! n - 1) - (x - 1) ! P(n) = Probabilitas dapat ‘success’ ke x dalam trial ke n n = # trial sampai dengan ‘sukses’ ke x p = Probabilitas ‘sukses’ x = # ‘sukses’

Distribusi Poisson

Model Distribusi Probabilitas Diskrit Hiper- Binomial Binomial Poisson Geometrik Negatif

Poisson Distribution # peristiwa yang terjadi pada suatu saat Contoh Peristiwa per unit Contoh: Waktu, rentang, area, ruang Contoh # pelanggan datang dalam 20 menit # pemogokkan setiap tahun # barang rusak diantara barang yang di pesan Other Examples: Number of machines that break down in a day Number of units sold in a week Number of people arriving at a bank teller per hour Number of telephone calls to customer support per hour

Proses Poisson Constant event probability One event per interval Average of 60/hr. is 1/min. for 60 1-minute intervals One event per interval Don’t arrive together Independent events Arrival of 1 person does not affect another’s arrival © 1984-1994 T/Maker Co.

Fungsi Distribusi Probabilitas Poisson P(X) = Probabilitas X (# ‘sukses’) l = Rerata # ‘sukses’ e = 2.71828 x = # ‘sukses’ per unit

Distribusi Karakteristik Poisson l = 0.5 Rerata l = 6 Deviasi Standar

Contoh Distribusi Poisson Customers arrive at a rate of 72 per hour. What is the probability of 4 customers arriving in 3 minutes? © 1995 Corel Corp.

Penyelesaian Distribusi Poisson 72 per hr. = 1.2 per min. = 3.6 per 3 min. interval

Soal You work in Quality Assurance for an investment firm. A clerk enters 75 words per minute with 6 errors per hour. What is the probability of 0 errors in a 255-word bond transaction?

Penyelesaian Distribusi Poisson: Mencari l* 75 words/min = (75 words/min)(60 min/hr) = 4500 words/hr 6 errors/hr = 6 errors/4500 words = .00133 errors/word In a 255-word transaction (interval): l = (.00133 errors/word )(255 words) = .34 errors/255-word transaction

Penyelesaian Distribusi Poisson: Mencari P(0)* X x ( ) ! . = e .7118 - 34 l

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