Beberapa Fungsi Peluang Kontinu (2) Pengantar Hitung Peluang - Pertemuan 10

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1 Beberapa Fungsi Peluang Kontinu (2) Pengantar Hitung Peluang - Pertemuan 10 r.rahma.anisa@gmail.com

2 Review Distribusi Seragam f ( x) b 1 0 a,untuk a x b,untuk x lainnya E( X ) a 2 b F X x = 0 x a (b a) 1, x < a, a x < b, x b V ( X ) ( b a) 12 2

3 Review - Distribusi Eksponensial

4 Aplikasi Distribusi Eksponensial Failure rate & reliabilitiy

5 Aplikasi Distribusi Eksponensial Failure rate & reliabilitiy

6 Ilustrasi - 1

7 Latihan - 1

8 Aplikasi Distribusi Eksponensial Modeling response time determine traffic intensity Queue lengths Average response time Pelajari lebih lanjut pada Horgan (2009)

9 Distribusi Normal Bentuk sebaran simetrik Mean, median dan modus berada dalam satu titik Fungsi kepekatan peluang dapat dituliskan sebagai berikut: 2 f ( x,, ) Peluang merupakan luasan dibawah kurva kepekatan normal: P ( - < x < + ) = P ( - 2 < x < + 2 ) = b 1 e 2 1 x 2 p ( a x b) f ( x) dx F( b) F( a) a 2 Peubah acak (X) dengan mean ( ) dan ragam ( 2 ) menyebar normal sering dituliskan sebagai X ~ N (, 2 )

10 Peubah Acak Normal

11 Peubah Acak Normal Baku

12 Tabel Distribusi Normal Baku

13 Ilustrasi - 2

14 Hampiran Normal untuk Sebaran Peluang Binomial Untuk ukuran contoh n yang besar, dan peluang p yang tidak mendekati 1 atau 0, sebaran peluang peubah acak binomial dapat didekati dengan sebaran normal dengan μ = np dan σ 2 = np(1 p). Pendekatan ini dapat digunakan hanya jika np 5 dan n(1 p) 5.

15 Continuity Correction Pada kasus dimana n tidak terlalu besar, koreksi sebesar ±0.5 dari nilai binomial akan meningkatkan presisi peluang yang didekati dengan sebaran normal. Ilustrasi pendekatan sebaran normal pada sebaran binomial

16 Ilustrasi - 3 Sebuah perkebunan teh mengklaim bahwa 20% hasil panennya merupakan daun teh dengan kualitas yang sangat tinggi. Jika pada suatu musim panen diperoleh 100 ton daun teh, berapakah peluang bahwa paling tidak terdapat 15 ton daun teh berkualitas tinggi? (Anggap asumsi binomial terpenuhi) μ = np = = 20 σ 2 = np 1 p = = 16 σ = 16 = 4

17 Hampiran Normal untuk Sebaran Peluang Poisson Apabila rata-rata sebaran Poisson lebih dari 10, maka pendekatan normal kepada binomial dapat diperluas kepada sebaran Poisson. Nilai harapan: E X = μ poisson = λ Ragam: 2 Var X = σ poisson = λ

18 Ilustrasi - 4 Rata-rata jumlah kendaraan yang mengunjungi bengkel pada jam di akhir pekan adalah 16 kendaraan. Berapakah peluang bahwa kurang dari 20 kendaraan akan mengunjungi bengkel pada jam yang sama di hari Selasaa mendatang?

19 Ilustrasi - 4 E X = μ poisson = λ = 16 2 Var X = σ poisson = λ = 16 σ = 16 = 4 Maka kasus ini dapat diselesaikan dengan cara: P X < 20 = P X 19.5 = P Z =

20 Distribusi Gamma α, β The Situation In the previous lesson, we learned that in an approximate Poisson process with mean λ, the waiting time X until the first event occurs follows an exponential distribution with mean β = 1/λ. We now let W denote the waiting time until the α th event occurs and find the distribution of W. We could represent the situation as follows:

21 Distribusi Gamma α, β α Note: β = 1 λ

22 Fungsi Gamma Γ α = dengan: Γ 1 = 0 0 y α 1 e y dy e y dy = 1 Γ α = α 1 Γ α 1 = α 1!

23 Kasus Khusus Distribusi Gamma Gamma 1, λ = Exponential(λ) Gamma α, 1 2 = Chi square(2α) Gamma α+β, 1 =Beta(α,β)

24 Ilustrasi - 4 Engineers designing the next generation of space shuttles plan to include two fuel pumps one active, the other in reserve. If the primary pump malfunctions, the second is automatically brought on line. Suppose a typical mission is expected to require that fuel be pumped for at most 50 hours. According to the manufacturer's specifications, pumps are expected to fail once every 100 hours. What are the chances that such a fuel pump system would not remain functioning for the full 50 hours?

25 Ilustrasi - 4 Solution. We are given that λ, the average number of failures in a 100-hour interval is 1. Therefore, β, the mean waiting time until the first failure is 1/λ, or 100 hours. Knowing that, let's now let Y denote the time elapsed until the α = 2nd pump breaks down. Assuming the failures follow a Poisson process, the probability density function of Y is: f y = 1 Γ α β α yα 1 e y f y = 1 y ye β = y 50 1 P Y < 50 = ye dy 1 Γ y2 1 e y 100, untuk y 0 (can be calculated using integral by parts, u=y, dv = e y 100 )

26 Ilustrasi 5 Total Compilation Time Compilation of a computer program consists of 3 blocks that are processed sequentially, one after another. Each block takes Exponential time with the mean of 5 minutes, independently of other blocks. a) Compute the expectation and variance of the total compilation time. b) Compute the probability for the entire program to be compiled in less than 12 minutes.

27 Ilustrasi - 5

28 Ilustrasi - 5

29 Latihan - 2 Lifetimes of computer memory chips have Gamma distribution with expectation μ = 12 years and standard deviation σ = 4 years. What is the probability that such a chip has a lifetime between 8 and 10 years?

30 Latihan - 3 For some electronic component, the time until failure has Gamma distribution with parameters α = 2 and λ = 2 (years 1). Compute the probability that the component fails within the first 6 months.

31 Latihan - 4 On the average, a computer experiences breakdowns every 5 months. The time until the first breakdown and the times between any two consecutive breakdowns are independent Exponential random variables. After the third breakdown, a computer requires a special maintenance. a) Compute the probability that a special maintenance is required within the next 9 months. b) Given that a special maintenance was not required during the first 12 months, what is the probability that it will not be required within the next 4 months?

32 Referensi 1. Baron, M Probability and Statistics for Computer Scientist, Second Edition. Boca Raton: CRC Press Taylor & Francis Group. 2. [Department of Statistics Online Programs] A Gamma Example. The Pennsylvania State University. [27 November 2016] 3. Horgan, J.M Probability with R: An Introduction with Computer Science Applications. New Jersey: John Wiley & Sons. 4. Montgomery, D.C, Runger, G.C Applied Statistics and Probability for Engineers, Third Edition. New Jersey: John Wiley & Sons. 5. Referensi lain yang relevan.