Problem: Hazard rate and density function for the gamma distribution

Let T be an gamma distributed stochastic variable with survival function given as \[S(t) = \frac{\Gamma(k,\gamma t)}{\Gamma(k)}, t>0,\] where \[\Gamma(k,x)= \int_x^\infty u^{k-1} e^{-u} du\] is the (upper) incomplete gamma function and \[\Gamma(k) = \int_0^\infty u^{k-1}e^{-u}du\] is the gamma function. Find formulas for the corresponding density function and hazard rate. Use also R to make plots of the density function, the hazard rate and the survival function when \(k=1\) and \(\gamma=1.5\), and when \(k=\frac{1}{2}\) and \(\gamma=\frac{1}{2}.\)

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