It’s an inspector’s characteristic to be absent-minded! (8)
I believe the answer is:
distrait
'absent-minded' is the definition.
(thesaurus)
'it's an inspector's characteristic' is the wordplay.
'it's an inspector's' becomes 'dis' (I've seen this in other clues).
'characteristic' becomes 'trait' (I've seen this before).
'dis'+'trait'='DISTRAIT'
'to be' acts as a link.
(Other definitions for distrait that I've seen before include "Being one [SCATTY]" , "Unable to concentrate for anxiety - it is trad (anag)" , "Preoccupied" , "Distraught in France" , "Inattentive or absent-minded, from French" .)