Pembahasan Ujian PAI: A20 – No. 14 – November 2017

PEMBAHASAN

\(E[X|X \ge 0] = \int_0^\infty {x\left( {\frac{{{f_X}(x)}}{{{\mathop{\rm P}\nolimits} (X \ge 0)}}} \right)} dx\) \(E[X|X \ge 0] = \int_0^\infty {x\left( {\frac{{\frac{1}{{\sqrt {2\pi } }}{e^{ – \frac{{{x^2}}}{2}}}}}{{{\mathop{\rm P}\nolimits} (X \ge 0)}}} \right)} dx\) \(E[X|X \ge 0] = \frac{1}{{{\mathop{\rm P}\nolimits} (X \ge 0)}}\frac{1}{{\sqrt {2\pi } }}\int_0^\infty {xe – } \frac{{{x^2}}}{2}dx\) \(E[X|X \ge 0] = \frac{1}{{0,5}}\frac{1}{{\sqrt {2\pi } }}\int_0^\infty {xe – } \frac{{{x^2}}}{x}dx\) \(subsitusi:\) \(y = – \frac{{{x^2}}}{2};dy = xdx\) \(E[X|X \ge 0] = \frac{2}{{\sqrt {2\pi } }}\int_0^\infty {x{e^{ – y}}} \frac{{dy}}{x}\) \(E[X|X \ge 0] = \frac{2}{{\sqrt {2\pi } }}\int_0^\infty {{e^{ – y}}} dy\) \(E[X|X \ge 0] = \frac{2}{{\sqrt {2\pi } }}(1)\) \(E[X|X \ge 0] = \sqrt {\frac{2}{\pi }}\)

Jawaban pada pilihan : B.\(\sqrt {\frac{2}{\pi }} \)