Karakteristik Contoh Acak · 2019-11-25 · Teori Statistika I (STK501) Dr. Kusman Sadik, M.Si...
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Teori Statistika I (STK501)
Dr. Kusman Sadik, M.Si
Program Studi Magister Statistika
Departemen Statistika IPB, 2019/2020
IPB University─ Bogor Indonesia ─ Inspiring Innovation with Integrity
Karakteristik Contoh AcakProperties of a Random Sample
(Bagian I)
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Contoh Acak
Salah satu aspek penting dalam statistika adalah
pengumpulan data (data collecting).
Ada dua metode pengumpulan data : (1).Perancangan
percobaan (experimental design) dan (2).Survei (suvey).
Pada kedua metode tersebut akan diperoleh segugus
data sebagai contoh (sample) dari populasi.
Salah satu karakteristik penting yang perlu dimiliki oleh
suatu contoh adalah acak (random).
Contoh yang memiliki karakteristik seperti itu disebut
sebagai contoh acak (random sample).
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Definisi
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Metode Penarikan Contoh
1. Dengan Pengembalian (wir)
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2. Tanpa Pengembalian (wor)
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Contoh Kasus (1)
( )
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( )
Ini menunjukkan bahwa untuk N besar, nilai pendekatan (approximation)
hampir sama dengan nilai yang sebenarnya (exactly).
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Statistik dan Sebaran Peluangnya
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Tiga Statistik yang Banyak Dipakai
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Notasi Statistik
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Nilai Harapan dan Ragam Contoh Acak
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Nilai Harapan dan Ragam Contoh Acak
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Nilai Harapan dan Ragam Contoh Acak
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Fungsi Pembangkit Momen dari Suatu Statistik
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Metode Konvolusi (Convolution)
Jika mencari sebaran dari ҧ𝑥 tidak berhasil melalui fungsi pembangkit momen
seperti di atas, maka dapat diselesaikan melalui metode konvolusi, yaitu melalui
proses transformasi peubah acak.
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Contoh Kasus (2):
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Penarikan Contoh dari Sebaran Normal
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Misalkan p.a. kontinu X menyebar Normal(0, 1) yaitu
fX(x) = 2
2
1
2
1 x
e
, - < x <
Jika didefinisikan p.a. Y = X2, ingin diketahui fkp bagi Y yaitu
fY(y).
Perhatikan bahwa dalam transformasi di atas, Y = X2, bukan fungsi satu-satu (one-to-one). Sehingga transformasi tersebut harus dipecah dulu agar menjadi fungsi satu-satu, yaitu: - < x 0 dan 0 < x < .
Contoh Kasus (3): Hubungan Sebaran Normal dan Khai-Kuadrat
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Untuk - < x 0
Y = h(X) = X2 X = h-1(Y) = Y
dan karena - < x 0 maka 0 y < .
J = dy
ydh )(1
= 2/12/1
2
1
2
1 yyydy
d
2/2/1
2/12
111
22
1
2
1.
2
1)())(()(
2
y
y
XY
ey
yedy
ydhyhfyf
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Untuk 0 < x <
Y = h(X) = X2 X = h-1(Y) = Y
dan karena 0 < x < maka 0 < y < .
J = dy
ydh )(1
= 2/12/1
2
1
2
1 yyydy
d
2/2/1
2/12
111
22
1
2
1.
2
1)())(()(
2
y
y
XY
ey
yedy
ydhyhfyf
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Sehingga fkp bagi p.a. Y adalah
0 ,2
1
22
1
22
1)(
2/2/1
2/2/12/2/1
yey
eyeyyf
y
yy
Y
Perhatikan bahwa fkp p.a. Y tersebut merupakan sebaran
Khai-Kuadrat dengan derajat bebas 1 yaitu 2(1).
Jadi jika X N(0, 1) maka Y = X2 2(1).
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Catatan : sebaran Khai-Kuadrat dengan derajat bebas r dapat
dinyatakan sebagai berikut:
0 ,2)2/(
1)( 2/1)2/(
2/
yey
ryf yr
r
untuk r = 1 maka (r/2) = , sehingga
0 ,2.
1
2
1)( 2/1)2/1(2/2/1 yeyeyyf yy
0 ,2)(
1 2/1)2/1(
2/12
1
yey y
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Karakteristik Sebaran Khai-Kuadrat
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Materi Responsi
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Materi Responsi (1)
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Materi Responsi (2)
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Materi Responsi (3)
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Materi Responsi (4)
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Pustaka
1. Casella, B. and R.L. Berger. 2002. Statistical Inference,
2nd Edition. Duxbury.
2. Hogg, R., Mc Kean, and Craig, A. 2005. Introduction to
Mathematical Statistics, 6th Edition. Prentice Hall.
3. Pustaka lain yang relevan.
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Catatan Kuliah
Bisa di-download di
kusmansadik.wordpress.com
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Terima Kasih