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Gemeinsame Verteilung und Grenzwertsätze
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4. Gemeinsame Verteilung und Grenzwertsätze
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4. Gemeinsame Verteilung und GrenzwertsätzeHäufig in der Praxis:
• Man muss mehrere (n) ZV’en gleichzeitig betrachten
(vgl. Statistik I, Kapitel 6)
Zunächst Vereinfachung:
• Betrachte n = 2 Zufallsvariablen (X und Y )
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,Beispiele:
• Zufällig ausgewählter Haushalt:
X = Haushaltsgröße
Y = Anzahl Autos
• Tagesrenditen zweier Aktien:
X = Rendite der VW-Aktie
Y = Rendite der BASF-Aktie
• 2-facher Würfelwurf:
X = Minimum der Augenzahlen
Y = Maximum der Augenzahlen
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,4.1 Gemeinsame Verteilung von Zufallsvariablen
Situation:
• Betrachte zwei ZV’en X und Y zu ein und demselben Zufalls-
experiment, d.h.
X : Ω −→ R
Y : Ω −→ R
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, Definition 4.1: (Gemeinsame Verteilungsfunktion)
Für die beiden ZV’en X und Y heißt die Funktion
FX,Y : R2 −→ [0, 1]
mit
FX,Y (x, y) = P ({ω : X(ω) ≤ x und Y (ω) ≤ y})
= P (X ≤ x, Y ≤ y)
die gemeinsame Verteilungsfunktion von X und Y .
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