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Diketahui fungsi peluang f(x) sebagai berikut. f(x)={kx+2​; untuk 2≤x≤40; untuk x yang lain​  d. Tentukan nilai peluang P(X≤3)! e. Tentukan nilai peluang P(X>2)!

Pertanyaan

Diketahui fungsi peluang f open parentheses x close parentheses sebagai berikut.

f open parentheses x close parentheses equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator x plus 2 over denominator k end fraction semicolon space text untuk end text space 2 less or equal than x less or equal than 4 end cell row cell 0 semicolon space text untuk end text space x space text yang lain end text end cell end table close 

d. Tentukan nilai peluang text P end text open parentheses X less or equal than 3 close parentheses!

e. Tentukan nilai peluang text P end text open parentheses X greater than 2 close parentheses!

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

 nilai peluang text P end text open parentheses X greater than 2 close parentheses adalah 1.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut berturut-turut adalah Error converting from MathML to accessible text. dan Error converting from MathML to accessible text..

Terlebih dahulu kita tentukan nilai k.

Fungsi f open parentheses x close parentheses equals fraction numerator x plus 2 over denominator k end fraction pada interval 2 less or equal than x less or equal than 4 merupakan fungsi peluang jika luas daerah di bawahnya sama dengan 1, yaitu :

table attributes columnalign right center left columnspacing 0px end attributes row cell integral subscript 2 superscript 4 f open parentheses x close parentheses d x end cell equals 1 row cell integral subscript 2 superscript 4 open parentheses fraction numerator x plus 2 over denominator k end fraction close parentheses d x end cell equals 1 row cell integral subscript 2 superscript 4 open parentheses 1 over k x plus 2 over k close parentheses d x end cell equals 1 row cell open square brackets fraction numerator x squared over denominator 2 k end fraction plus fraction numerator 2 x over denominator k end fraction close square brackets subscript 2 superscript 4 end cell equals 1 row cell open parentheses fraction numerator 4 squared over denominator 2 k end fraction plus fraction numerator 2 open parentheses 4 close parentheses over denominator k end fraction close parentheses minus open parentheses fraction numerator 2 squared over denominator 2 k end fraction plus fraction numerator 2 open parentheses 2 close parentheses over denominator k end fraction close parentheses end cell equals 1 row cell fraction numerator 16 over denominator 2 k end fraction plus 8 over k minus fraction numerator 4 over denominator 2 k end fraction minus 4 over k end cell equals 1 row cell fraction numerator 12 over denominator 2 k end fraction plus 4 over k end cell equals 1 row cell 6 over k plus 4 over k end cell equals 1 row cell 10 over k end cell equals 1 row k equals 10 end table 

Maka diperoleh untuk interval 2 less or equal than x less or equal than 4 fungsi f open parentheses x close parentheses equals fraction numerator x plus 2 over denominator 10 end fraction.

Soal nomor d.

Peluang terambil nilai data pada interval c less or equal than X less or equal than d sama dengan luas daerah dibawah kurva pada interval tersebut. Dapat dituliskan :

straight P open parentheses c less or equal than X less or equal than d close parentheses equals integral subscript straight c superscript straight d straight f open parentheses straight x close parentheses d straight x 

Menentukan nilai peluang text P end text open parentheses X less or equal than 3 close parentheses yaitu :

table attributes columnalign right center left columnspacing 0px end attributes row cell text P end text open parentheses X less or equal than 3 close parentheses end cell equals cell P open parentheses 2 less than x less or equal than 3 close parentheses end cell row blank equals cell integral subscript 2 superscript 3 f open parentheses x close parentheses d x end cell row blank equals cell integral subscript 2 superscript 3 open parentheses fraction numerator x plus 2 over denominator 10 end fraction close parentheses d x end cell row blank equals cell 1 over 10 integral subscript 2 superscript 3 open parentheses x plus 2 close parentheses d x end cell row blank equals cell 1 over 10 open square brackets 1 half x squared plus 2 x close square brackets subscript 2 superscript 3 end cell row blank equals cell 1 over 10 open parentheses 3 squared over 2 plus 2 open parentheses 3 close parentheses minus open parentheses 2 squared over 2 plus 2 open parentheses 2 close parentheses close parentheses close parentheses end cell row blank equals cell 1 over 10 open parentheses 9 over 2 plus 6 minus 4 over 2 minus 4 close parentheses end cell row blank equals cell 1 over 10 open parentheses 5 over 2 plus 2 close parentheses end cell row blank equals cell 1 over 10 open parentheses 5 over 2 plus 4 over 2 close parentheses end cell row blank equals cell 1 over 10 open parentheses 9 over 2 close parentheses end cell row blank equals cell 9 over 20 end cell end table 

Dengan demikian, nilai peluang text P end text open parentheses X less or equal than 3 close parentheses adalah 9 over 20.

Soal nomor e.

Menentukan nilai peluang text P end text open parentheses X greater than 2 close parentheses yaitu :

table attributes columnalign right center left columnspacing 0px end attributes row cell text P end text open parentheses X greater than 2 close parentheses end cell equals cell P open parentheses 2 less than X less or equal than 4 close parentheses end cell row blank equals cell integral subscript 2 superscript 4 f open parentheses x close parentheses d x end cell row blank equals cell integral subscript 2 superscript 4 open parentheses fraction numerator x plus 2 over denominator 10 end fraction close parentheses d x end cell row blank equals cell 1 over 10 integral subscript 2 superscript 4 open parentheses x plus 2 close parentheses d x end cell row blank equals cell 1 over 10 open square brackets 1 half x squared plus 2 x close square brackets subscript 2 superscript 4 end cell row blank equals cell 1 over 10 open parentheses 4 squared over 2 plus 2 open parentheses 4 close parentheses minus open parentheses 2 squared over 2 plus 2 open parentheses 2 close parentheses close parentheses close parentheses end cell row blank equals cell 1 over 10 open parentheses 16 over 2 plus 8 minus open parentheses 4 over 2 plus 4 close parentheses close parentheses end cell row blank equals cell 1 over 10 open parentheses 8 plus 8 minus open parentheses 2 plus 4 close parentheses close parentheses end cell row blank equals cell 1 over 10 open parentheses 16 minus 6 close parentheses end cell row blank equals cell 1 over 10 open parentheses 10 close parentheses end cell row blank equals 1 end table 

Dengan demikian, nilai peluang text P end text open parentheses X greater than 2 close parentheses adalah 1.

96

4.0 (7 rating)

Nova Khoirun Nisa

Pembahasan tidak menjawab soal

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Diketahui fungsi peluang f(x) sebagai berikut. f(x)={kx+2​; untuk 2≤x≤40; untuk x yang lain​  a. Tentukan nilai k! b. Tentukan nilai peluang P(2≤X≤3)! c. Tentukan nilai peluang P(3≤X≤4)!

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