Dichtefunktion Beispiel Essay

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Beispiel: Lars experimentiert mit seiner Kugelbahn. Die über mehrere Kurven und verschiedene Hindernisse abwärts geführte Kugel rollt auf einer waagerechten Schiene aus. Lars stellt fest, dass sie bei den mehrfach durchgeführten Versuchen an verschiedenen Stellen zum Liegen kommt. Die Entfernung vom Beginn der waagerechten Rollschiene bis zu dem Punkt, an dem die Kugel anhält, ist eine Zufallsgröße X. Die Kugel rollt jedes Mal mindestens 150 cm, aber höchstens 250 cm weit, d.h., die Zufallsgröße X kann alle Werte aus dem Intervall [150; 250] annehmen.

Eine Zufallsgröße, die prinzipiell jeden Wert aus einer überabzählbaren Menge annehmen kann (sei es aus einem beschränkten Intervall oder aus der Menge ), nennt man im Allgemeinen eine stetige Zufallsgröße. Dabei muss man beachten, dass der Begriff „stetige Zufallsgröße“ von dem Begriff der „stetigen Funktion“ (wie er in der Analysis verwandt wird) abweicht. Mit der Bezeichnung einer Zufallsgröße als stetig ist man lediglich darauf aus, dass die Wertemenge von X nicht diskret ist, sondern ein Kontinuum bildet. Mitunter wird deshalb auch von einer zufälligen reellen Zahl gesprochen.

Kehren wir nun zu obigem Beispiel zurück.
Beim Bestimmen der Wahrscheinlichkeitsverteilung von X stößt Lars auf eine Schwierigkeit. Die Werte von X bilden eine überabzählbar unendliche Menge und deshalb gilt , denn kann keine von null verschiedene Zahl sein, da sonst die Summe aller Wahrscheinlichkeiten nicht gleich 1 wäre.

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