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Pertanyaan

Diketahui fungsi kepadatan variabel acak kontinu X didefinisikan sebagai berikut untuk 0 < x < 1. Jika ,maka Var(X) = ....

Diketahui fungsi kepadatan variabel acak kontinu X didefinisikan sebagai berikut

begin mathsize 14px style f open parentheses x close parentheses equals a x plus b x squared end style

untuk 0 < x < 1. Jika begin mathsize 14px style E open parentheses X squared close parentheses equals 11 over 20 end style, maka Var(X) = ....

  1. begin mathsize 14px style 19 over 120 end style 

  2. begin mathsize 14px style fraction numerator 487 over denominator 1.200 end fraction end style 

  3. fraction numerator size 14px 139 over denominator size 14px 2.880 end fraction 

  4. fraction numerator size 14px 739 over denominator size 14px 6.400 end fraction 

  5. fraction numerator size 14px 2.869 over denominator size 14px 14.400 end fraction 

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A. Khairunisa

Master Teacher

Mahasiswa/Alumni Universitas Negeri Semarang

Jawaban terverifikasi

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Pembahasan

Perhatikan bahwa X adalah variabel acak kontinu yang nilainya berada di antara 0 dan 1. Sehingga Kemudian diketahui bahwa . Sehingga Perhatikan bahwa didapat dua persamaan, yaitu Eliminasi kedua persamaan tersebut. Sehingga didapat Sehingga Maka Selanjutnya didapat bahwa Sehingga

Perhatikan bahwa X adalah variabel acak kontinu yang nilainya berada di antara 0 dan 1. Sehingga

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript 0 superscript 1 f open parentheses x close parentheses d x end cell equals 1 row cell integral subscript 0 superscript 1 open parentheses a x plus b x squared close parentheses d x end cell equals 1 row cell open a over 2 x squared plus b over 3 x cubed close square brackets subscript 0 superscript 1 end cell equals 1 row cell open parentheses a over 2 open parentheses 1 close parentheses squared plus b over 3 open parentheses 1 close parentheses cubed close parentheses minus open parentheses a over 2 open parentheses 0 close parentheses squared plus b over 3 open parentheses 0 close parentheses close parentheses end cell equals 1 row cell open parentheses a over 2 plus b over 3 close parentheses minus open parentheses 0 plus 0 close parentheses end cell equals 1 row cell a over 2 plus b over 3 end cell equals 1 end table end style

Kemudian diketahui bahwa begin mathsize 14px style E open parentheses X squared close parentheses equals 11 over 20 end style. Sehingga

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell E open parentheses X squared close parentheses end cell equals cell 11 over 20 end cell row cell integral subscript 0 superscript 1 x squared times f open parentheses x close parentheses d x end cell equals cell 11 over 20 end cell row cell integral subscript 0 superscript 1 x squared open parentheses a x plus b x squared close parentheses d x end cell equals cell 11 over 20 end cell row cell integral subscript 0 superscript 1 open parentheses a x cubed plus b x to the power of 4 close parentheses d x end cell equals cell 11 over 20 end cell row cell open a over 4 x to the power of 4 plus b over 5 x to the power of 5 close square brackets subscript 0 superscript 1 end cell equals cell 11 over 20 end cell row cell open parentheses a over 4 open parentheses 1 close parentheses to the power of 4 plus b over 5 open parentheses 1 close parentheses to the power of 5 close parentheses minus open parentheses a over 4 open parentheses 0 close parentheses to the power of 4 plus b over 5 open parentheses 0 close parentheses to the power of 5 close parentheses end cell equals cell 11 over 20 end cell row cell open parentheses a over 4 plus b over 5 close parentheses minus open parentheses 0 plus 0 close parentheses end cell equals cell 11 over 20 end cell row cell a over 4 plus b over 5 end cell equals cell 11 over 20 end cell row cell 2 open parentheses a over 4 plus b over 5 close parentheses end cell equals cell 2 open parentheses 11 over 20 close parentheses end cell row cell a over 2 plus fraction numerator 2 b over denominator 5 end fraction end cell equals cell 11 over 10 end cell end table end style

Perhatikan bahwa didapat dua persamaan, yaitu

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell a over 2 plus b over 3 end cell equals 1 row cell a over 2 plus fraction numerator 2 b over denominator 5 end fraction end cell equals cell 11 over 10 end cell end table end style

Eliminasi kedua persamaan tersebut. Sehingga didapat

begin mathsize 14px style table row cell a over 2 plus b over 3 equals 1 end cell blank row cell a over 2 plus fraction numerator 2 b over denominator 5 end fraction equals 11 over 10 end cell minus row cell b over 3 minus fraction numerator 2 b over denominator 5 end fraction equals negative 1 over 10 end cell blank row cell fraction numerator 5 b over denominator 15 end fraction minus fraction numerator 6 b over denominator 15 end fraction equals negative 1 over 10 end cell blank row cell negative b over 15 equals negative 1 over 10 end cell blank row cell b equals negative 1 over 10 open parentheses negative 15 close parentheses end cell blank row cell b equals 3 over 2 end cell blank end table end style

Sehingga

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell a over 2 plus b over 3 end cell equals 1 row cell a over 2 plus fraction numerator open parentheses 3 over 2 close parentheses over denominator 3 end fraction end cell equals 1 row cell a over 2 plus 1 half end cell equals 1 row cell a over 2 end cell equals cell 1 half end cell row a equals 1 end table end style

Maka

begin mathsize 14px style f open parentheses x close parentheses equals a x plus b x squared f open parentheses x close parentheses equals x plus 3 over 2 x squared end style

Selanjutnya didapat bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row mu equals cell E open parentheses X close parentheses end cell row blank equals cell integral subscript 0 superscript 1 x times f open parentheses x close parentheses d x end cell row blank equals cell integral subscript 0 superscript 1 x open parentheses x plus 3 over 2 x squared close parentheses d x end cell row blank equals cell integral subscript 0 superscript 1 open parentheses x squared plus 3 over 2 x cubed close parentheses d x end cell row blank equals cell open 1 third x cubed plus 3 over 8 x to the power of 4 close square brackets subscript 0 superscript 1 end cell row blank equals cell open parentheses 1 third open parentheses 1 close parentheses cubed plus 3 over 8 open parentheses 1 close parentheses to the power of 4 close parentheses minus open parentheses 1 third open parentheses 0 close parentheses cubed plus 3 over 8 open parentheses 0 close parentheses to the power of 4 close parentheses end cell row blank equals cell open parentheses 1 third plus 3 over 8 close parentheses minus open parentheses 0 plus 0 close parentheses end cell row blank equals cell 1 third plus 3 over 8 end cell row blank equals cell 8 over 24 plus 9 over 24 end cell row blank equals cell 17 over 24 end cell end table end style

Sehingga

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell V a r open parentheses X close parentheses end cell equals cell E open parentheses X squared close parentheses minus mu squared end cell row blank equals cell 11 over 20 minus open parentheses 17 over 24 close parentheses squared end cell row blank equals cell 11 over 20 minus 289 over 576 end cell row blank equals cell fraction numerator 1.584 over denominator 2.880 end fraction minus fraction numerator 1.445 over denominator 2.880 end fraction end cell row blank equals cell fraction numerator 139 over denominator 2.880 end fraction end cell end table end style 

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