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Pertanyaan

Diketahui X adalah variabel acak kontinu yang nilainya berada di antara -3 dan 4. Jika fungsi kepadatan dari X adalah f(x) = k|x|, maka Var(X) = ....

Diketahui X adalah variabel acak kontinu yang nilainya berada di antara -3 dan 4. Jika fungsi kepadatan dari X adalah f(x) = k|x|, maka Var(X) = ....

  1. begin mathsize 14px style 863 over 150 end style 

  2. fraction numerator size 14px 5.913 over denominator size 14px 2.500 end fraction 

  3. fraction numerator size 14px 6.041 over denominator size 14px 7.500 end fraction 

  4. fraction numerator size 14px 64.873 over denominator size 14px 11.250 end fraction 

  5. fraction numerator size 14px 744.769 over denominator size 14px 22.500 end fraction 

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A. Khairunisa

Master Teacher

Mahasiswa/Alumni Universitas Negeri Semarang

Jawaban terverifikasi

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Pembahasan

Dari soal di atas dapat diketahui bahwa batas bawah dari variabel acak X adalah -3 dan batas atasnya adalah 4. Sehingga Ingat bahwa Karena nilai-nilai dari X berada di antara -3 dan 4, maka Maka Sehingga Kemudian didapat bahwa Sehingga

Dari soal di atas dapat diketahui bahwa batas bawah dari variabel acak X adalah -3 dan batas atasnya adalah 4. Sehingga

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript negative 3 end subscript superscript 4 f open parentheses x close parentheses d x end cell equals 1 row cell integral subscript negative 3 end subscript superscript 4 k open vertical bar x close vertical bar d x end cell equals 1 end table end style

Ingat bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar x close vertical bar end cell equals cell open curly brackets table attributes columnalign right end attributes row cell x comma end cell cell x greater or equal than 0 end cell row cell negative x comma end cell cell x less than 0 end cell end table close end cell row cell k open vertical bar x close vertical bar end cell equals cell open curly brackets table attributes columnalign right end attributes row cell k x comma end cell cell x greater or equal than 0 end cell row cell negative k x comma end cell cell x less than 0 end cell end table close end cell end table end style

Karena nilai-nilai dari X berada di antara -3 dan 4, maka

begin mathsize 14px style f open parentheses x close parentheses equals k open vertical bar x close vertical bar equals open curly brackets table attributes columnalign right end attributes row cell k x comma end cell cell 0 less or equal than x less than 4 end cell row cell negative k x comma end cell cell negative 3 less than x less than 0 end cell end table close end style

Maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript negative 3 end subscript superscript 4 k open vertical bar x close vertical bar d x end cell equals 1 row cell integral subscript negative 3 end subscript superscript 0 open parentheses negative k x close parentheses d x plus integral subscript 0 superscript 4 k x d x end cell equals 1 row cell open negative k over 2 x squared close square brackets subscript negative 3 end subscript superscript 0 plus open k over 2 x squared close square brackets subscript 0 superscript 4 end cell equals 1 row cell open parentheses negative k over 2 open parentheses 0 close parentheses squared minus open parentheses negative k over 2 open parentheses negative 3 close parentheses squared close parentheses close parentheses plus open parentheses k over 2 open parentheses 4 close parentheses squared minus k over 2 open parentheses 0 close parentheses squared close parentheses end cell equals 1 row cell open parentheses 0 minus open parentheses negative fraction numerator 9 k over denominator 2 end fraction close parentheses close parentheses plus open parentheses 8 k minus 0 close parentheses end cell equals 1 row cell fraction numerator 9 k over denominator 2 end fraction plus 8 k end cell equals 1 row cell fraction numerator 25 k over denominator 2 end fraction end cell equals 1 row k equals cell 2 over 25 end cell end table end style

Sehingga

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 negative 3 end subscript superscript 4 x times f open parentheses x close parentheses d x end cell row blank equals cell integral subscript negative 3 end subscript superscript 4 x times k open vertical bar x close vertical bar d x end cell row blank equals cell integral subscript negative 3 end subscript superscript 0 x open parentheses negative k x close parentheses d x plus integral subscript 0 superscript 4 x open parentheses k x close parentheses d x end cell row blank equals cell integral subscript negative 3 end subscript superscript 0 open parentheses negative k x squared close parentheses d x plus integral subscript 0 superscript 4 k x squared d x end cell row blank equals cell integral subscript negative 3 end subscript superscript 0 open parentheses negative 2 over 25 x squared close parentheses d x plus integral subscript 0 superscript 4 2 over 25 x squared d x end cell row blank equals cell open negative 2 over 75 x cubed close square brackets subscript negative 3 end subscript superscript 0 plus open 2 over 75 x cubed close square brackets subscript 0 superscript 4 end cell row blank equals cell open parentheses negative 2 over 75 open parentheses 0 close parentheses cubed minus open parentheses negative 2 over 75 open parentheses negative 3 close parentheses cubed close parentheses close parentheses plus open parentheses 2 over 75 open parentheses 4 close parentheses cubed minus 2 over 75 open parentheses 0 close parentheses cubed close parentheses end cell row blank equals cell open parentheses 0 minus 18 over 25 close parentheses plus open parentheses 128 over 75 minus 0 close parentheses end cell row blank equals cell negative 18 over 25 plus 128 over 75 end cell row blank equals cell negative 54 over 75 plus 128 over 75 end cell row blank equals cell 74 over 75 end cell end table end style

Kemudian didapat bahwa

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 integral subscript negative 3 end subscript superscript 4 x squared times f open parentheses x close parentheses d x end cell row blank equals cell integral subscript negative 3 end subscript superscript 4 x squared times k open vertical bar x close vertical bar d x end cell row blank equals cell integral subscript negative 3 end subscript superscript 0 x squared open parentheses negative k x close parentheses d x plus integral subscript 0 superscript 4 x squared open parentheses k x close parentheses d x end cell row blank equals cell integral subscript negative 3 end subscript superscript 0 open parentheses negative k x cubed close parentheses d x plus integral subscript 0 superscript 4 k x cubed d x end cell row blank equals cell integral subscript negative 3 end subscript superscript 0 open parentheses negative 2 over 25 x cubed close parentheses d x plus integral subscript 0 superscript 4 2 over 25 x cubed d x end cell row blank equals cell open negative 1 over 50 x to the power of 4 close square brackets subscript negative 3 end subscript superscript 0 plus open 1 over 50 x to the power of 4 close square brackets subscript 0 superscript 4 end cell row blank equals cell open parentheses negative 1 over 50 open parentheses 0 close parentheses to the power of 4 minus open parentheses negative 1 over 50 open parentheses negative 3 close parentheses to the power of 4 close parentheses close parentheses plus open parentheses 1 over 50 open parentheses 4 close parentheses to the power of 4 minus 1 over 50 open parentheses 0 close parentheses cubed close parentheses end cell row blank equals cell open parentheses 0 minus open parentheses negative 81 over 50 close parentheses close parentheses plus open parentheses 256 over 50 minus 0 close parentheses end cell row blank equals cell 81 over 50 plus 256 over 50 end cell row blank equals cell 337 over 50 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 337 over 50 minus open parentheses 74 over 75 close parentheses squared end cell row blank equals cell 337 over 50 minus fraction numerator 5.476 over denominator 5.625 end fraction end cell row blank equals cell fraction numerator 75.825 over denominator 11.250 end fraction minus fraction numerator 10.952 over denominator 11.250 end fraction end cell row blank equals cell fraction numerator 64.873 over denominator 11.250 end fraction end cell end table end style 

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Diketahui X adalah variabel acak kontinu yang nilainya berada di antara 0 dan 4. Jika fungsi kepadatan dari X adalah ,maka Var(X) = ....

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