Diketahui
Fungsi survival:
\[ S(t)=\frac{100-t}{100}, \quad 0 \le t \le 100 \]
(a) Probabilitas usia 20 hidup sampai usia 70
Probabilitas seseorang yang sudah hidup sampai usia 20 akan hidup sampai usia 70 adalah:
\[ P(T>70 \mid T>20)=\frac{S(70)}{S(20)} \]
Hitung:
\[ S(70)=\frac{100-70}{100}=\frac{30}{100}=0.3 \]
\[ S(20)=\frac{100-20}{100}=\frac{80}{100}=0.8 \]
Maka:
\[ P(T>70 \mid T>20)=\frac{0.3}{0.8}=0.375=37.5\% \]
(b) Laju kematian pada usia 50
Rumus laju kematian:
\[ \mu(t) = -\frac{d}{dt}\ln(S(t)) \]
Dengan:
\[ S(t)=\frac{100-t}{100} \]
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:
\[ \mu(50)=\frac{1}{100-50}=\frac{1}{50}=0.02 \]
(c) Harapan hidup lengkap pada usia 20
Rumus harapan hidup lengkap:
\[ e_x=\int_0^\infty {}_tp_x \, dt \]
Dengan:
\[ {}_tp_x=\frac{S(x+t)}{S(x)} \]
Maka:
\[ e_{20}=\int_0^{80}\frac{S(20+t)}{S(20)}dt \]
Karena:
\[ S(20+t)=\frac{100-(20+t)}{100}=\frac{80-t}{100} \]
dan:
\[ S(20)=\frac{80}{100}=0.8 \]
Sehingga:
\[ \frac{S(20+t)}{S(20)} =\frac{\frac{80-t}{100}}{\frac{80}{100}} =\frac{80-t}{80} \]
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:
\[ e_{20}=40 \text{ tahun} \]
# Fungsi survival
S <- function(t){
(100 - t)/100
}
# (a)
p_20_to_70 <- S(70) / S(20)
# (b)
mu <- function(t){
1/(100 - t)
}
mu_50 <- mu(50)
# (c)
integrand <- function(x){
S(20 + x) / S(20)
}
e_20 <- integrate(integrand, lower = 0, upper = 80)$value
# Output
p_20_to_70## [1] 0.375mu_50## [1] 0.02e_20## [1] 40Kesimpulan
- Probabilitas usia 20 hidup sampai 70:
\[ 0.375 = 37.5\% \]
- Laju kematian pada usia 50:
\[ 0.02 \]
- Harapan hidup lengkap pada usia 20:
\[ 40 \text{ tahun} \]
```