Kuadrat dari simpangan baku adalah varian atau ragam. Varians digunakan untuk mengetahui seberapa jauh persebaran nilai hasil observasi terhadap rata – rata. Varians merupakan ukuran penyeba…Full description
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Variansi dan Kovariansi Baha Bahan n Kuli Kuliah ah II20 II2092 92 Prob Probab abililit itas as dan dan Stat Statis isti tik k Oleh: Oleh: Rinald Rinaldii Munir Munir Sekolah Teknik Elektro dan Informatika ITB
1
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Variansi • Kita sudah memahami bahwa nilai harapan peubah acak X se seringkali dise isebut rataan (mean) dan dan dila dilamb mban angk gkan an dengan μ. • Tetapi, rataan tidak memberikan gambaran dispersi atau pencar ncara an data data.. Rata Rataan an dari ari masi masing ng-m -ma asing sing peuba eubah h acak acak berb berbed eda a mung mungki kin n sama sama,, mesk meskip ipun un dist distri ribu busi siny nya a tida tidak k sama. ama. Ol Oleh eh kare arena itu itu dip diperlu erluk kan besa besara ran n lain lain yang yang meng mengga gamb mbar arka kan n seba sebara ran n data data..
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Definisi . Misalkan X adal adalah ah vari variab abel el rand random om deng dengan an distribusi peluang f(X) dan rataan μ. Var Varia ians nsii dari dari X adalah:
jika X diskrit, dan
jika X kontinu.
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• Interpretasi : Nilai x – μ dise disebu butt peny penyim impa pang ngan an suat suatu u peng pengam amat atan an dari dari rata rataan anny nya. a. Kare Karena na peny penyim impa pang ngan an ini ini diku dikuad adra ratk tkan an lalu lalu dira dirata ta-r -rat atak akan an,, maka maka σ2 akan leb lebih kecil untuk kelompok nilai lai x ya yang de dekat μ dibandingkan dengan kelom lompok nila ilai x ya yang ja jauh dari μ. • Dengan kata lain, jika nilai-nilai x cenderng terk terko onsen nsentr tras asii di deka dekatt rata rataa annya, nya, mak maka varia arians nsin inya ya kecil. il. Se Sedangkan jik jika jau jauh dari rataan maka variansinya besar.
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Vari Varian ansi si keci kecill
Vari Varian ansi si besa besar r
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Contoh 1. Dib Diberik rikan disr disrib ibus usii pelua eluang ng sbb: bb: x f(x)
1 0.3
Hitun itungl glah ah vari varia ansi nsi dari ari X. Jawaban:
2 0.4
3 0.3
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• Variansi juga dapat dihitung dengan rumus lain yang yang leb lebih ih muda mudah, h, yai yaitu: tu: σ2
= E(X2) –
μ2
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• Latihan . Sebu Sebuah ah pan panitia itia bera eranggo nggota taka kan n 3 ora orang dipil ipilih ih secara acak dari 4 or orang mahasiswa STI da dan 3 or orang maha mahasi sisw swa a IF. IF. Hitu Hitung ng vari varian ansi siny nya. a. (jawaban ada pada slid lide beriku ikut ini)
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•
Contoh ont oh 3. Misa Misalk lkan an X meny menyat atak akan an perm permin inta taan an miny minyak ak gore goreng ng (dal (dalam am liter iter)) menje menjela lang ng hari hari raya raya.. Fun Fungsi gsi pada padatt dari dari X sebagai berikut:
Cari rataan dan variansi X.
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Variansi untuk peubah acak lain ya yang be bergantung pada X, yaitu g(X), diberikan dala teorema di bawah ini. Teorema. Misalkan X adalah peubah acak dengan dist distrib ribus usii pelu peluan ang g f(x). Variansi dari peubah acak g(X)
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Contoh 4 Hitu Hitu
lah lah
ri
si dari dari g(X) = 2X + 3, bila X
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Contoh 5. Mi
lk
X = j mlah mlah ballp allp int int w
bi , d
Y
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