How to Solve Insurance Risk Variables — Univariate distributions — identify a model from its assumptions Questions on Exam P
Sample Practice Problem
ID: #FAE1G
A proposed CDF is \(F(x)=0\) for \(x\leq0\), \(F(x)=cx^{1}\) for \(0<x<2\), and \(F(x)=1\) for \(x\geq2\). Calculate the constant \(c\) that makes \(F\) a valid CDF.
📖 Worked Solution & Strategy
A cumulative distribution function must be nondecreasing, right-continuous, and approach one. Matching the interior formula to the upper branch determines its constant, after which differentiation or inversion answers the other requests. For \(F(x)\) to be a valid continuous CDF, right-continuity at the upper boundary \(x=2\) requires \(F(2)=c(2)^{1}=1\). Solving yields \(c=2^{-1}=\frac{1}{2}\).