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:

$(t)=c{t}^{-1}+(1-c)t^{-1}{e}^{t^{}}$

For the given special case of equations, plot the curve for $f(t)$ and $(t).$ Explain the significance of the curves in each case using

practical $/$ real$-$life examples.

Exponential $($i$.$e$.,$ for $c=1,=1)$

Rayleigh $($i$.$e$.,$ for $c=1,=2)$

Weibull $($i$.$e$.,$ for $c=1)$

Extreme value $($i$.$e$.,$ for $c=0,=1)$

Makeham $($i$.$e$.,$ for $=1,=1)$