Non-monotone Survival Function?

Question

I was given the task of plotting the graph of a survival function with the following details defined.

• hazardrate (lambda) = λ(t) = 0.2 (1 + sin(tπ/12))
• the underlying process is exponential

I am calculating the h(t) for each period and it basically equals to lambda based on what we have learnt in case of an exponential PDF.

It gives me a cyclic hazardrate but using the S(t) = exp(-lambda*t) leaves me a non monontone decreasing function.

I am pretty sure that I am approaching the issue from a wrong perspective and kindly asking you to help me out! Many thanks in advance!

Answer 1

You have hazard rate

$$ \lambda(t) = 0.2(1+\sin(t\pi/12)) $$

The survival function $S(y)$ -- the proportion surviving through time $y$ -- is

$$ S(y) = \exp\bigg(-\int_0^y \lambda(t) dt\bigg) \\ $$

Computing $S(y)$ is an exercise in the basic rules of integration. Since $\lambda(t)\geq 0$, we know that $S(y)$ is monotone non-increasing in $y$.