Problem: Survival and density functions for the Weibull distribution

Let T be a Weibull distributed stochastic variable. The hazard rate of \(T\) is then given as \[\alpha(t)=b t^{k-1}, t>0.\] Find formulas for the corresponding density and survival functions. Use also R to make plots of the density function, the hazard rate and the survival function when \(b=1\) and \(k=2\), and when \(b=\frac{1}{2}\) and \(k=\frac{1}{2}.\)

Hints: To get some hints for how to solve the problem see the link at the bottom of this page.