Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Distribusi masa hidup Jaka, , sebuah peubah acak eksponensial dengan mean ; dan distribusi masa hidup Dedi, , sebuah peubah acak eksponensial dengan mean ; Masa hidup Jaka dan Dedi saling bebas. Cari peluang Jaka hidup lebih lama dibanding Dedi!
- \(\frac{\alpha }{{\alpha + \beta }}\)
- \(\frac{\beta }{{\alpha + \beta }}\)
- \(\frac{{\alpha – \beta }}{\alpha }\)
- \(\frac{{\beta – \alpha }}{\beta }\)
- \(\frac{\alpha }{\beta }\)
\(P(X > Y) = \int\limits_0^\infty {\int\limits_0^x {f(x,y)} f(x,y)dydx} \)
\(P(X > Y) = \int\limits_0^\infty {\int\limits_0^x {(\frac{1}{{\alpha \beta }}{e^{ – (\frac{x}{\alpha } + \frac{y}{\beta })}})dydx} } \)
\(P(X > Y) = \frac{{ – \beta }}{{\alpha \beta }}\int\limits_0^\infty {({e^{ – (\frac{x}{\alpha })}})({e^{^{ – (\frac{x}{\beta })}}} – 1)dx} \)
\(P(X > Y) = \frac{1}{\alpha }\int\limits_0^\infty {({e^{ – (\frac{x}{\alpha })}} – {e^{ – x(\frac{1}{\alpha } + \frac{1}{\beta })}})dx} \)
\(P(X > Y) = \frac{1}{\alpha }[( – \alpha )(0 – 1) – ( – (\frac{1}{{\frac{1}{\alpha } + \frac{1}{\beta }}}))(0 – 1)]\)
\(P(X > Y) = \frac{1}{\alpha }[\alpha – (\frac{{\alpha \beta }}{{\alpha + \beta }})]\)
\(P(X > Y) = 1 – (\frac{\beta }{{\alpha + \beta }})\)
\(P(X > Y) = \frac{{\alpha + \beta – \beta }}{{\alpha + \beta }}\)
\(P(X > Y) = \frac{\alpha }{{\alpha + \beta }}\)