Pertanyaan

Diketahui X adalah variabel acak kontinu yang nilainya berada di antara 6 dan 10. Jika fungsi kepadatan dari X adalah maka Var(X) = ....

A. Khairunisa

Master Teacher

Mahasiswa/Alumni Universitas Negeri Semarang

Jawaban terverifikasi

Pembahasan

Dari soal di atas dapat diketahui bahwa batas bawah dari variabel acak X adalah 6 dan batas atasnya adalah 10. Sehingga Kemudian didapat pula bahwa Sehingga

Dari soal di atas dapat diketahui bahwa batas bawah dari variabel acak X adalah 6 dan batas atasnya adalah 10. Sehingga

$begin mathsize 14px style mu equals E open parentheses X close parentheses equals integral subscript 6 superscript 10 x times f open parentheses x close parentheses d x equals integral subscript 6 superscript 10 x times 3 over 16 open parentheses x minus 8 close parentheses squared d x equals integral subscript 6 superscript 10 fraction numerator 3 x over denominator 16 end fraction open parentheses x squared minus 16 x plus 64 close parentheses d x equals integral subscript 6 superscript 10 fraction numerator 3 x cubed over denominator 16 end fraction minus 3 x squared plus 12 x d x equals open fraction numerator 3 x to the power of 4 over denominator 64 end fraction minus x cubed plus 6 x squared close square brackets subscript 6 superscript 10 equals open x squared open parentheses fraction numerator 3 x squared over denominator 64 end fraction minus x plus 6 close parentheses close square brackets subscript 6 superscript 10 equals 10 squared open parentheses fraction numerator 3 open parentheses 10 close parentheses squared over denominator 64 end fraction minus 10 plus 6 close parentheses minus 6 squared open parentheses fraction numerator 3 open parentheses 6 close parentheses squared over denominator 64 end fraction minus 6 plus 6 close parentheses equals 100 open parentheses 300 over 64 minus 4 close parentheses minus 36 open parentheses 108 over 64 close parentheses equals 100 open parentheses 300 over 64 minus 256 over 64 close parentheses minus 36 open parentheses 108 over 64 close parentheses equals 100 open parentheses 44 over 64 close parentheses minus 36 open parentheses 108 over 64 close parentheses equals 100 open parentheses 11 over 16 close parentheses minus 36 open parentheses 27 over 16 close parentheses equals 25 open parentheses 11 over 4 close parentheses minus 9 open parentheses 27 over 4 close parentheses equals 275 over 4 minus 243 over 4 equals 32 over 4 equals 8 end style$

Kemudian didapat pula bahwa

$begin mathsize 14px style E open parentheses X squared close parentheses equals integral subscript 6 superscript 10 x squared times f open parentheses x close parentheses d x equals integral subscript 6 superscript 10 x squared times 3 over 16 open parentheses x minus 8 close parentheses squared d x equals integral subscript 6 superscript 10 fraction numerator 3 x squared over denominator 16 end fraction open parentheses x squared minus 16 x plus 64 close parentheses d x equals integral subscript 6 superscript 10 fraction numerator 3 x to the power of 4 over denominator 16 end fraction minus 3 x cubed plus 12 x squared d x equals open fraction numerator 3 x to the power of 5 over denominator 80 end fraction minus fraction numerator 3 x to the power of 4 over denominator 4 end fraction plus 4 x cubed close square brackets subscript 6 superscript 10 equals open x cubed open parentheses fraction numerator 3 x squared over denominator 80 end fraction minus fraction numerator 3 x over denominator 4 end fraction plus 4 close parentheses close square brackets subscript 6 superscript 10 equals 10 cubed open parentheses fraction numerator 3 open parentheses 10 close parentheses squared over denominator 80 end fraction minus fraction numerator 3 open parentheses 10 close parentheses over denominator 4 end fraction plus 4 close parentheses minus 6 cubed open parentheses fraction numerator 3 open parentheses 6 close parentheses squared over denominator 80 end fraction minus fraction numerator 3 open parentheses 6 close parentheses over denominator 4 end fraction plus 4 close parentheses equals 1.000 open parentheses 300 over 80 minus 30 over 4 plus 4 close parentheses minus 216 open parentheses 108 over 80 minus 18 over 4 plus 4 close parentheses equals 1.000 open parentheses 300 over 80 minus 600 over 80 plus 320 over 80 close parentheses minus 216 open parentheses 108 over 80 minus 360 over 80 plus 320 over 80 close parentheses equals 1.000 open parentheses 20 over 80 close parentheses minus 216 open parentheses 68 over 80 close parentheses equals 1.000 open parentheses 1 fourth close parentheses minus 216 open parentheses 17 over 20 close parentheses equals 250 minus 54 open parentheses 17 over 5 close parentheses equals fraction numerator 1.250 over denominator 5 end fraction minus 918 over 5 equals 332 over 5 end style$

Sehingga