A300 ENCOUNTERS UNEXPECTED WX RELATED TURB BTWN LIMA AND MIA. PAX IN RESTROOM; WITH SEATBELT SIGN ON; IS INJURED.

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INITIAL REVIEW OF FLT PAPERWORK INDICATES TURB INDEX 0 FOR ENTIRE FLT ROUTING. INITIALLY CLRED TO FL280 DUE TO LIMA AIR TFC. FO MADE PA ANNOUNCEMENT IN BOTH ENGLISH AND SPANISH ADVISING PAX WHILE IN SEATS PLEASE KEEP SEATBELTS FASTENED WHILE SEATED AND THAT FLC IS NOT ALWAYS ABLE TO GIVE ADVANCED WARNING OF UNANTICIPATED TURB. APPROX 45 MINS ENRTE ENCOUNTERED MODERATE CHOP. TURNED ON SEATBELT SIGN; ADVISED CABIN CREW TO TERMINATE SVC AND BE SEATED. ADVISED FLT SUPPORT LIMA AND DISPATCH OF RIDE CONDITIONS. SOLICITED RIDE RPTS FROM LIMA CTR AND REQUESTED FL310. CLRED TO FL310. 10 MINS LATER; RIDE IMPROVED. FLT ATTENDANTS RESUMED SVC; TURNED OFF SEATBELT SIGN. BTWN POINTS ARNAL AND GRAND CAYMEN; WE WERE DEVIATING AROUND RAINSHOWERS AS A PRECAUTION. ILLUMINATED SEATBELT SIGN. IN AND OUT OF HIGH STRATUS. ENCOUNTERED VMC AND NOTICED A BUILDUP NOT PAINTING ON RADAR. FO INITIATED A TURN TO THE R TO AVOID BUILDUP. AT THIS TIME; ENCOUNTERED UNEXPECTED MODERATE TURB. FO DISCONNECTED AUTOPLT; ALLOWING ACFT TO RIDE THE TURB -- NOT APPLYING OPPOSING CTL FORCES TO AGGRAVATE THE SIT. FLT ATTENDANTS ADVISED THAT DURING THIS TIME; WHILE THE SEATBELT SIGN WAS ILLUMINATED; ONE PAX DECIDED TO USE THE RESTROOM AND WAS INJURED. FLT LANDED AT MIA.