AN ACFT APCHING LGA EXECUTES A MISSED APCH. ACTUAL WX IS BETTER THAN STATED ON ATIS WHICH THE FLC SEES DURING THE GAR.

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WHILE ON VECTORS TO ILS DME RWY 13 AT LGA; WE FLEW 4000 FT JUST W OF LGA. I COULD SEE ENTIRE ARPT AND IMPROVING WX CONDITIONS OVER PESSIMISTIC ATIS. AFTER LONG VECTORS (10 MINS AT LEAST); WE WERE CLRED FOR APCH. DURING FINAL SEGMENT; WITH FIELD IN SIGHT; TWR ISSUED RVR 1000 FT TOUCHDOWN; 1200 FT ROLLOUT. WE CONTINUED TO 400-450 FT MSL AND CALLED MISSED APCH. WHILE ON GS I COULD SEE ENTIRE ARPT. RWY 13 FIRST 2000 FT AND LAST 3000 FT WERE VISIBLE WITH FOG COVERING MID SECTION. DURING VECTORS AFTER MISSED APCH ASKED FOR PIREP. I TOLD WHAT I SAW AND RPTED RADAR ECHOES IN AREA. CAPT WAS OFF RADIO WITH COMPANY AND PAX. WITH CAPT BACK ON; WE AGREED WITH APCH CTL TO MAKE ANOTHER ATTEMPT. PRIOR TO APCH WE RECEIVED NO NEW WX. THE SECOND ATTEMPT WAS ALMOST IDENTICAL TO FIRST IN EXECUTION AND VISUAL CONTACTS WITH ARPT AND RWY. ON FINAL SEGMENT; RVR WAS AGAIN RPTED BY TWR AS BELOW MINIMUMS. MISSED APCH WAS EXECUTED AT 400-450 FT MSL AND WE PROCEEDED TO ALTERNATE. I WAS NOT CONCERNED WITH RVR BECAUSE THE WX WAS IMPROVING. THE RESULT WAS NOT THINKING TO CHK WX PRIOR TO SECOND ATTEMPT. I PERCEIVED BOTH APCHS WOULD BE SUCCESSFUL AND JUDGED IT SAFE TO MAKE SECOND ATTEMPT. WE SHOULD HAVE MADE A BETTER ATTEMPT TO GET ACCURATE WX PRIOR TO SECOND ATTEMPT. WE COULD HAVE ASKED FOR AN APCH TO ANOTHER RWY. MY CHK-IN FOR THE TRIP THAT WAS XY35. I WOKE AT XX30 AFTER RETIRING THE PREVIOUS NIGHT AT YZ45. THE PREVIOUS NIGHT I SLEPT FROM AA15 TO FF30. DAY #2 AND THE NIGHT OF DAY #1; I HAD FLU SYMPTOMS THAT HAD DISAPPEARED BY DAY #3. MY SLEEP THOSE 2 NIGHTS WAS SOMEWHAT DISTURBED. BY THE TIME I WAS FLYING THOSE ATTEMPTED 2 APCHS; I WAS TIRED. I WAS FLYING; BUT NOT PLANNING AS WELL AS I SHOULD HAVE.