Pertanyaan

Diketahui X adalah variabel acak kontinu yang nilainya berada di antara 5 dan 8. Jika fungsi kepadatan dari X adalah , maka μ = ....

Diketahui X adalah variabel acak kontinu yang nilainya berada di antara 5 dan 8. Jika fungsi kepadatan dari X adalah $begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator open parentheses x minus 4 close parentheses open parentheses 10 minus x close parentheses over denominator 24 end fraction end style$ , maka μ = ....

$begin mathsize 14px style fraction numerator 3.533 over denominator 80 end fraction end style$
$begin mathsize 14px style fraction numerator 1.989 over denominator 288 end fraction end style$

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 5 dan batas atasnya adalah 8. Sehingga

$begin mathsize 14px style mu equals E open parentheses X close parentheses space space equals integral subscript 5 superscript 8 x times f open parentheses x close parentheses d x space space equals integral subscript 5 superscript 8 x times fraction numerator open parentheses x minus 4 close parentheses open parentheses 10 minus x close parentheses over denominator 24 end fraction d x space space equals integral subscript 5 superscript 8 x times fraction numerator negative x squared plus 14 x minus 40 over denominator 24 end fraction d x space space equals integral subscript 5 superscript 8 fraction numerator negative x cubed plus 14 x squared minus 40 x over denominator 24 end fraction d x space space equals integral subscript 5 superscript 8 minus x cubed over 24 plus fraction numerator 7 x squared over denominator 12 end fraction minus fraction numerator 5 x over denominator 3 end fraction d x space space equals open negative x to the power of 4 over 96 plus fraction numerator 7 x cubed over denominator 36 end fraction minus fraction numerator 5 x squared over denominator 6 end fraction close square brackets subscript 5 superscript 8 space space equals open parentheses negative open parentheses 8 close parentheses to the power of 4 over 96 plus fraction numerator 7 open parentheses 8 close parentheses cubed over denominator 36 end fraction minus fraction numerator 5 open parentheses 8 close parentheses squared over denominator 6 end fraction close parentheses minus open parentheses negative open parentheses 5 close parentheses to the power of 4 over 96 plus fraction numerator 7 open parentheses 5 close parentheses cubed over denominator 36 end fraction minus fraction numerator 5 open parentheses 5 close parentheses squared over denominator 6 end fraction close parentheses space space equals 8 squared open parentheses negative open parentheses 8 close parentheses squared over 96 plus fraction numerator 7 open parentheses 8 close parentheses over denominator 36 end fraction minus 5 over 6 close parentheses minus 5 squared open parentheses negative open parentheses 5 close parentheses squared over 96 plus fraction numerator 7 open parentheses 5 close parentheses over denominator 36 end fraction minus 5 over 6 close parentheses space space equals 64 open parentheses negative 64 over 96 plus 56 over 36 minus 5 over 6 close parentheses minus 25 open parentheses negative 25 over 96 plus 35 over 36 minus 5 over 6 close parentheses space space equals 64 open parentheses negative 2 over 3 plus 14 over 9 minus 5 over 6 close parentheses minus 25 open parentheses negative 25 over 96 plus 35 over 36 minus 30 over 36 close parentheses space space equals 64 open parentheses negative 12 over 18 plus 28 over 18 minus 15 over 18 close parentheses minus 25 open parentheses negative 25 over 96 plus 5 over 36 close parentheses space space equals 64 open parentheses 1 over 18 close parentheses minus 25 open parentheses negative 75 over 288 plus 40 over 288 close parentheses space space equals 64 open parentheses 1 over 18 close parentheses minus 25 open parentheses negative 35 over 288 close parentheses space space equals 64 over 18 plus 875 over 288 space space equals fraction numerator 1.024 over denominator 288 end fraction plus 875 over 288 space space equals fraction numerator 1.899 over denominator 288 end fraction space space equals 211 over 32 end style$