Tugas Fungsi Survival

Diketahui

Fungsi survival:

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

---

(a) Probabilitas usia 20 hidup sampai usia 70

Rumus:

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

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 dan Harapan Hidup

Laju Kematian pada usia 50

Rumus:

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

Karena:

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

Maka:

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

Sehingga:

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

---

Harapan Hidup Lengkap Usia 20

Rumus:

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

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

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

\[ = \frac{3200}{80} \]

\[ = 40 \]

---

(c) Kesimpulan

\[ P(T > 70 \mid T > 20) = 0.375 = 37.5\% \]

Pernyataan tersebut benar.

## Warning: package 'dplyr' was built under R version 4.5.2

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

## Warning: package 'ggplot2' was built under R version 4.5.2

## ===== HASIL PERHITUNGAN =====

## 1. Probabilitas usia 20 hidup sampai 70 = 0.375 atau 37.5 %

## 2. Laju kematian pada usia 50 = 0.02

## 3. Harapan hidup lengkap pada usia 20 = 40 tahun

## Artinya rata-rata hidup sampai usia = 60 tahun

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once per session.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.