Definition: Indikatorfunktion

Sei Ω MathType@MTEF@5@5@+= feaagKart1ev2aqaMPbvLHfij5gC1rhimfMBNvxyNvgaC9uBLDwyam XvP5wqSXMqHnxAJn0BKvguHDwzZbqefqvATv2CG4uz3bIuV1wyUbqe dmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8srps 0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr 0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaci GacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbiabfM6axbaa@40EF@ Grundraum und A⊆Ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGbbGaeyOHI0SaeuyQdCfaaa@3A6C@ ein Ereignis. Dann ist die Indikatorfunktion 1 A ( ω )={ 1 falls ω∈A 0 falls ω∉A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaWdamaaBaaaleaapeGaamyqaaWdaeqaaOWaaeWaaeaapeGa eqyYdChapaGaayjkaiaawMcaaiabg2da9maaceaabaqbaeqabiqaaa qaaiaaigdacaqGGaGaaeOzaiaabggacaqGSbGaaeiBaiaabohacaqG GaWdbiabeM8a3jabgIGiolaadgeaa8aabaGaaGimaiaabccacaqGMb GaaeyyaiaabYgacaqGSbGaae4CaiaabccapeGaeqyYdCNaeyycI8Sa amyqaaaaa8aacaGL7baaaaa@534E@ eine Zufallsvariable die angibt, ob das Ereignis eintritt oder nicht.

Eigenschaften der Indikatorfunktion

Seien A,B⊆Ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGbbGaaiilaiaadkeacqGHgksZcqqHPoWvaaa@3BE3@ zwei Ereignisse, so gilt

  1. 1 A ¯ =1− 1 A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaWaaSbaaSqaaiqadgeagaqeaaqabaGccqGH9aqpcaaIXaGa eyOeI0IaaGyma8aadaWgaaWcbaWdbiaadgeaa8aabeaaaaa@3C6F@
  2. 1 A∪B = 1 A + 1 B − 1 A∩B =max{ 1 A ,1 B } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaWaaSbaaSqaaiaadgeacqGHQicYcaWGcbaabeaakiabg2da 9iaaigdadaWgaaWcbaGaamyqaaqabaGccqGHRaWkcaaIXaWaaSbaaS qaaiaadkeaaeqaaOGaeyOeI0IaaGymamaaBaaaleaacaWGbbGaeyyk ICSaamOqaaqabaGccqGH9aqpciGGTbGaaiyyaiaacIhadaGadaqaai aaigdapaWaaSbaaSqaa8qacaWGbbaapaqabaGccaGGSaGaaGymamaa BaaaleaapeGaamOqaaWdaeqaaaGcpeGaay5Eaiaaw2haaaaa@4F1C@
  3. Für A⊆B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGbbGaeyOHI0SaamOqaaaa@39A5@ gilt 1 A ≤ 1 B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaaIXaWdamaaBaaaleaapeGaamyqaaWdaeqaaOWdbiabgsMiJkaa igdapaWaaSbaaSqaa8qacaWGcbaapaqabaaaaa@3B9D@ .