Problem: Given a probability density function $f(t)$ and required parameters,. Derive the Reliability and Hazard Rate expression. Show all intermediate steps and

equations to reach the final expression for the Hazard Rate $(t)$ based on the provided parameters.

$f(t)=[c{t}^{-1}+(1-c)t^{-1}{e}^{t^{}}][exp[-c{t}^{}-(1-c)(e^{t^{}}-1)]]$

for $0c1$ and $,,,>0$

where $t=$ time,

$$ and $=$ scale parameters

$$ and $=$ shape parameters

Final Expression: $,(t)=c{t}^{-1}+(1-c)t^{-1}{e}^{t^{}}$