Diketahui fungsi survival : \[S(t)=\frac{100-t}{100}, \quad 0 \le t \le 100\]

a) probabilitas seorang berusia 20 tahun hidup hingga usia 70

\[ P(T>70 \mid T>20)=\frac{S(70)}{S(20)} \] Hitung:

\[ S(70)=\frac{100-70}{100}=\frac{30}{100}=0.3 \]

\[ S(20)=\frac{100-20}{100}=\frac{80}{100}=0.8 \]

Sehingga :

\[ P(T>70 \mid T>20)=\frac{0.3}{0.8}=0.375=37.5\% \]

Rumus laju kematian:

\[ \mu(t) = -\frac{d}{dt}\ln(S(t)) \]

Dengan yang diketahui S(t) :

\[ S(t)=\frac{100-t}{100} \]

Maka:

\[ \ln(S(t))=\ln(100-t)-\ln(100) \]

Turunkan:

\[ \frac{d}{dt}\ln(S(t))=-\frac{1}{100-t} \]

Sehingga:

\[ \mu(t)=\frac{1}{100-t} \]

Nilai pada usia 50:

\[ \mu(50)=\frac{1}{100-50}=\frac{1}{50}=0.02 \]

Rumus :

\[ e_x=\int_0^\infty {}_tp_x \, dt \]

Dengan:

\[ {}_tp_x=\frac{S(x+t)}{S(x)} \]

Sehingga :

\[ e_{20}=\int_0^{80}\frac{S(20+t)}{S(20)}dt \]

Karena:

\[ S(20+t)=\frac{100-(20+t)}{100}=\frac{80-t}{100} \]

dan:

\[ S(20)=\frac{80}{100}=0.8 \]

Maka :

\[ \frac{S(20+t)}{S(20)} =\frac{\frac{80-t}{100}}{\frac{80}{100}} =\frac{80-t}{80} \]

Jadi:

\[ e_{20}=\int_0^{80}\frac{80-t}{80}\,dt \]

Hitung:

\[ e_{20}=\frac{1}{80}\int_0^{80}(80-t)\,dt \]

\[ =\frac{1}{80}\left[80t-\frac{t^2}{2}\right]_0^{80} \]

\[ =\frac{1}{80}(6400-3200)=40 \]

Jadi hasilnya:

\[ e_{20}=40 \text{ tahun} \]