I need to calculate the hazard rate of the Pareto distribution.
I know that the residual life distribution looks at the remaining waiting time given that you have already waited for a certain amount of time x, and the hazard rate can be thought of as the likelihood of the wait time ending now, given that you have waited x time already
But I don't quite see how to use that in order to compute it.
This is actually pretty easy to calculate. The definition of the hazard function is $f(x)/S(x)$, where $f(x)$ is the pdf and $S(x)$ is the survival function. The survival function is just defined as $S(x)=1-F(x)$, where $F(x)$ is the cdf. If we go with the Wikipedia parameterization of the Pareto distribution, then we get a hazard rate as follows: $$h(x)=f(x)/(1-F(x))=((\alpha x_m^\alpha)/x^{\alpha+1})/(1-(1-(x_m/x)^\alpha))=\alpha/x$$