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Stetige Zufallsvariable Dichtefunktion Beispiel Essay

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Stetige Zufallsvariable

In diesem Kapitel schauen wir uns an, was eine stetige Zufallsvariable ist.

Eine Funktion \(X\), die
jedem Ergebnis \(\omega\) des Ergebnisraum \(\Omega\)
genau eine Zahl \(x\) der Menge der reellen Zahlen \(\mathbb{R}\)
zuordnet, heißt Zufallsvariable.

Kurzschreibweise: \(X: \Omega \rightarrow \mathbb{R}\)

Diese Definition lässt sich in einem Mengendiagramm sehr leicht veranschaulichen.




Eine Zufallsvariable ordnet
jedem \(\omega_i\) aus \(\Omega\)
genau ein \(x_i\) aus \(\mathbb{R}\)
zu.

Lass dich von dem Wort Zufallsvariable nicht verwirren! Eine Zufallsvariable \(X\) ist keine Zahl, die in einem Zufallsexperiment zufällig herauskommt, sondern eine Funktion, die jedem zufällig entstehenden Ergebnis \(\omega\) einen ganz genau bestimmten Zahlenwert \(x\) zuordnet: \(X: \omega \rightarrow x\).
Merke: Die Werte \(x\), die eine Zufallsvariable \(X\) annimmt, heißen auch „Realisationen“.

Eine Zufallsvariable \(X\) wird als stetig bezeichnet,
wenn sie überabzählbar unendlich viele Werte annimmt.

Beispiele

\(X := \text{„Gewicht einer zufällig ausgewählten Person“}\)
           \(\Rightarrow\) unendliche Wertemenge, die nicht abzählbar ist

\(X := \text{„Geschwindigkeit eines an einer Radarkontrolle vorbeifahrenden Autos“}\)
           \(\Rightarrow\) unendliche Wertemenge, die nicht abzählbar ist

Stetige Zufallsvariablen entstehen meist durch einen Messvorgang.

Beim Messen von physikalischen Größen (wie Länge, Masse, Volumen, Temperatur, Zeit etc.) spielen viele kleine Störeinflüsse eine Rolle, die das Messergebnis mal etwas zu hoch, mal etwas zu niedrig ausfallen lassen. Unabhängig von der Messgenauigkeit kann eine stetige Zufallsvariable innerhalb eines Intervalls unendlich viele Werte annehmen.

Stetige Wahrscheinlichkeitsverteilung

Eine Wahrscheinlichkeitsverteilung gibt an,
wie sich die Wahrscheinlichkeiten
auf die möglichen Werte einer Zufallsvariablen verteilen.

Die Wahrscheinlichkeitsverteilung einer stetigen Zufallsvariablen lässt sich beschreiben durch:

Beispiel

Ein Zufallsgenerator erzeugt zufällig eine Zahl zwischen 2,5 und 4,5.

1.) Dichtefunktion

\(\begin{equation*} f(x) = \begin{cases} 0 & \text{für } x < 2,5\\ \frac{1}{2} & \text{für } 2,5 \le x \le 4,5 \\ 0 & \text{für } x > 4,5 \end{cases} \end{equation*}\)

Merke: \(f(x) \neq P(X = x)\)
Die Wahrscheinlichkeit, dass eine stetige Zufallsvariable \(X\) einen bestimmten Wert \(x\) annimmt, ist stets Null. Folglich gilt:
\(P(X = x) = 0\)

Aus der Dichtefunktion selbst lassen sich keine Wahrscheinlichkeiten ablesen.
Vielmehr gibt die Fläche unter der Dichtefunktion die Wahrscheinlichkeit an.

Bei stetigen Zufallsvariablen verwendet man deshalb zur Berechnung von Wahrscheinlichkeiten die entsprechende Verteilungsfunktion. Sie ergibt sich durch Integration der Dichtefunktion:

\[F(x) = P(X \le x) = \int_{-\infty}^{x} \! f(u) \, \mathrm{d}u\]

Die Dichtefunktion hat nur die Aufgabe, einen visuellen Eindruck der Verteilung zu vermitteln.

2.) Verteilungsfunktion

\(\begin{equation*} F(x) = \begin{cases} 0 & \text{für } x \le 2,5\\ \frac{1}{2}x-\frac{5}{4} & \text{für } 2,5 < x < 4,5 \\ 1 & \text{für } x \geq 4,5 \end{cases} \end{equation*}\)

Merke: \(F(x) = P(X \le x)\)

Sowohl die Dichtefunktion als auch die Verteilungsfunktion beschreiben die Wahrscheinlichkeitsverteilung einer stetigen Zufallsvariablen vollständig. Häufig ist eine vollständige Beschreibung der Verteilung gar nicht notwendig. Um sich einen groben Überblick über eine Verteilung zu verschaffen, betrachtet man einige charakteristische Maßzahlen.
Dazu zählen u.a. der Erwartungswert, die Varianz und die Standardabweichung.

Stetige Zufallsvariablen im Überblick

Entstehung durch...Messvorgang
Beispiel:Gewicht einer zufällig ausgewählten Person
Wahrscheinlichkeitsverteilung 
- Dichtefunktion 
- Verteilungsfunktion 
Maßzahlen 
- Erwartungswert\[\mu_{X} = \mathrm{E}(X) = \int_{-\infty}^{\infty} \! x \cdot f(x) \, \mathrm{d}x\]
- Varianz\[\sigma^2_{X} = \mathrm{Var}(X) = \int_{-\infty}^{\infty} \! (x - \mu_{X})^2 \cdot f(x) \, \mathrm{d}x\]
- Standardabweichung\[\sigma_{X} = \sqrt{\mathrm{Var}(X)}\]