Let $T$ be the survival time and $T$ follows an exponential distribution. Derive the expression for survival function of $T$. Also obtain expression for force of mortality.
Solution
Let $T$ be the survival time. The simplest parametric model for survival data is the exponential distribution with density function
$$ \begin{aligned} f(t) = \lambda e^{-\lambda t}, t> 0, \lambda >0. \end{aligned} $$
Distribution function of exponential dist.
The distribution function of $T$ is
$$ \begin{align} F(t) & = P(T
Thus the survival function is
$$ \begin{aligned} S(t) = 1-F(t) = e^{-\lambda t}. \end{aligned} $$
The force of mortality is given by
$$ \begin{aligned} \mu(t) & = \frac{f(t)}{1-F(t)}=-\frac{S^\prime(t)}{S(t)}\\ &= \frac{\lambda e^{-\lambda t}}{1-\big(1-e^{-\lambda t}\big)}\\ &=\frac{\lambda e^{-\lambda t}}{e^{-\lambda t}}\\ &= \lambda . \end{aligned} $$
OR
The force of mortality is given by
$$ \begin{aligned} \mu(t) & = -\frac{S^\prime(t)}{S(t)}\\ &= -\frac{-\lambda e^{-\lambda t}}{e^{-\lambda t}}\\ &= \lambda . \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
