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Jika variabel acak , nilai  adalah ...

Jika variabel acak begin mathsize 14px style straight Z tilde straight N open parentheses 0 comma 1 close parentheses end style, nilai begin mathsize 14px style straight P open parentheses straight Z less or equal than 3 close parentheses end style adalah ...

  1. begin mathsize 14px style integral subscript 0 superscript 3 fraction numerator 1 over denominator square root of 2 straight pi end root end fraction straight e to the power of negative 1 half straight z squared end exponent d straight z end style 

  2. begin mathsize 14px style integral subscript 3 superscript infinity fraction numerator 1 over denominator square root of 2 straight pi end root end fraction straight e to the power of negative 1 half straight z squared end exponent d straight z end style 

  3. begin mathsize 14px style integral subscript negative infinity end subscript superscript 3 fraction numerator 1 over denominator square root of 2 straight pi end root end fraction straight e to the power of negative 1 half straight z squared end exponent d straight z end style 

  4. begin mathsize 14px style integral subscript 0 superscript 3 fraction numerator 1 over denominator straight sigma square root of 2 straight pi end root end fraction straight e to the power of negative 1 half open parentheses fraction numerator straight x minus straight mu over denominator straight sigma end fraction close parentheses squared end exponent d straight z end style 

  5. begin mathsize 14px style integral subscript negative infinity end subscript superscript 3 fraction numerator 1 over denominator straight sigma square root of 2 straight pi end root end fraction straight e to the power of negative 1 half open parentheses fraction numerator straight x minus straight mu over denominator straight sigma end fraction close parentheses end exponent squared d straight z end style 

Jawaban:

Fungsi peluang variabel acak begin mathsize 14px style straight Z tilde straight N open parentheses 0 comma space 1 close parentheses end style didefinisikan sebagai:

begin mathsize 14px style f open parentheses straight z close parentheses equals straight P open parentheses straight Z equals straight z close parentheses equals fraction numerator 1 over denominator square root of 2 straight pi end root end fraction straight e to the power of negative 1 half straight z squared end exponent end style untuk begin mathsize 14px style negative infinity less than straight z less than infinity end style.

Fungsi distributif komulatif variabel acak begin mathsize 14px style straight Z tilde straight N open parentheses 0 comma space 1 close parentheses end style didefinisikan sebagai:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight F open parentheses straight z close parentheses end cell equals cell straight P open parentheses straight Z less or equal than straight z close parentheses equals integral subscript negative infinity end subscript superscript z f open parentheses straight z close parentheses d straight z equals fraction numerator 1 over denominator square root of 2 straight pi end root end fraction integral subscript negative infinity end subscript superscript straight z straight e to the power of negative 1 half straight z squared end exponent d straight z end cell end table end style.

Maka, untuk begin mathsize 14px style straight Z tilde straight N open parentheses 0 comma 1 close parentheses end style dengan nilai begin mathsize 14px style straight P open parentheses straight Z less or equal than 3 close parentheses end style adalah sebagai berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight P open parentheses straight Z less or equal than straight z close parentheses end cell equals cell integral subscript negative infinity end subscript superscript straight z f open parentheses straight z close parentheses d straight z end cell row cell straight P open parentheses straight Z less or equal than 3 close parentheses end cell equals cell integral subscript negative infinity end subscript superscript 3 fraction numerator 1 over denominator square root of 2 straight pi end root end fraction straight e to the power of negative 1 half straight z squared end exponent d straight z end cell row blank equals cell fraction numerator 1 over denominator square root of 2 straight pi end root end fraction integral subscript negative infinity end subscript superscript 3 straight e to the power of negative 1 half straight z squared end exponent dz end cell end table end style

Jadi, jawaban yang tepat adalah C.

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