Inverse-Transformation Method gives complex results

Question

Given the following pdf $$ f(x)=2x^{-3},\;\;\;1<x<\infty $$ it seems nature to me to use the inverse-transformation method. find that $$ F(x)=-x^{-2} $$ and set $$ x=-U^{-\frac{1}{2}} $$ where $U\sim(0,1)$. However, this arrangement means all the $x$'s are complex since we have taking square root of a negative number. Is there any thoughts on what is going wrong here? Or inverse-transformation is simply not applicable here?

Answer 1

Your cdf doesn't make sense, since it wants to calculate negative probabilities. Make sure you take the correct antiderivative $$\int_1^x f(t) \, \mathrm{d}t.$$