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

$$ \begin{aligned} \mu(t) & = -\frac{S^\prime(t)}{S(t)}\\ &= -\frac{-\lambda e^{-\lambda t}}{e^{-\lambda t}}\\ &= \lambda . \end{aligned} $$