Dreiecksverteilung: Unterschied zwischen den Versionen
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Kowa (Diskussion | Beiträge) Keine Bearbeitungszusammenfassung |
Kowa (Diskussion | Beiträge) Keine Bearbeitungszusammenfassung |
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pdf_image =| | pdf_image =| | ||
cdf_image =| | cdf_image =| | ||
parameters =<math>a \in ]-\infty,\infty[</math><br><math>b \in ]-\infty,\infty[,\,b>a</math><br><math>c \in ]a,b[</math><br><math>m := \frac{c-a}{b-a},\,1-m=\frac{b-c}{b-a}</math>| | parameters =<math>a \in ]-\infty,\infty[</math><br><math>b \in ]-\infty,\infty[,\,b>a</math><br><math>c \in ]a,b[</math><br><math>m := \frac{c-a}{b-a},\,1-m=\frac{b-c}{b-a},\,c = a+m(b-a)</math>| | ||
support =<math>]a,b[ \!</math>| | support =<math>]a,b[ \!</math>| | ||
pdf =<math> | pdf =<math> | ||
f(x) = | f(x) = | ||
\begin{cases} | \begin{cases} | ||
\frac{2(x-a)}{(b-a)(c-a)}, & \mbox{wenn } a \le x \le c \\ | \frac{2(x-a)}{(b-a)(c-a)} = \frac{2(x-a)}{m(b-a)^2}, & \mbox{wenn } a \le x \le c \\ | ||
\frac{2(b-x)}{(b-a)(b-c)}, & \mbox{wenn } c < x \le b \\ | \frac{2(b-x)}{(b-a)(b-c)} = \frac{2(x-a)}{(1-m)(b-a)^2}, & \mbox{wenn } c < x \le b \\ | ||
0, & \mbox{sonst } | 0, & \mbox{sonst } | ||
\end{cases} | \end{cases} | ||
| Zeile 17: | Zeile 17: | ||
F(x) = | F(x) = | ||
\begin{cases} | \begin{cases} | ||
0, | 0, & \mbox{wenn } x < a\\ | ||
\frac{(x-a)^2}{(b-a)(c-a)}, | 0+\frac{(x-a)^2}{(b-a)(c-a)} = 0+\frac{(x-a)^2}{m(b-a)^2}, & \mbox{wenn } a \le x \le c \\ | ||
1-\frac{(b-x)^2}{(b-a)(b-c)}, & \mbox{wenn } c < x \le b \\ | 1-\frac{(b-x)^2}{(b-a)(b-c)} = 1-\frac{(b-x)^2}{(1-m)(b-a)^2}, & \mbox{wenn } c < x \le b \\ | ||
1, | 1, & \mbox{wenn } b < x | ||
\end{cases} | \end{cases} | ||
</math>| | </math>| | ||
Version vom 26. Mai 2006, 18:20 Uhr
| Parameter | $ a \in ]-\infty,\infty[ $ $ b \in ]-\infty,\infty[,\,b>a $ $ c \in ]a,b[ $ $ m := \frac{c-a}{b-a},\,1-m=\frac{b-c}{b-a},\,c = a+m(b-a) $ |
| Dichtefunktion | $ f(x) = \begin{cases} \frac{2(x-a)}{(b-a)(c-a)} = \frac{2(x-a)}{m(b-a)^2}, & \mbox{wenn } a \le x \le c \\ \frac{2(b-x)}{(b-a)(b-c)} = \frac{2(x-a)}{(1-m)(b-a)^2}, & \mbox{wenn } c < x \le b \\ 0, & \mbox{sonst } \end{cases} $ |
| Träger | $ ]a,b[ \! $ |
| Verteilungsfunktion | $ F(x) = \begin{cases} 0, & \mbox{wenn } x < a\\ 0+\frac{(x-a)^2}{(b-a)(c-a)} = 0+\frac{(x-a)^2}{m(b-a)^2}, & \mbox{wenn } a \le x \le c \\ 1-\frac{(b-x)^2}{(b-a)(b-c)} = 1-\frac{(b-x)^2}{(1-m)(b-a)^2}, & \mbox{wenn } c < x \le b \\ 1, & \mbox{wenn } b < x \end{cases} $ |
| Modus | $ c\, $ |
| Erwartungswert | $ \mu = \frac{a+b+c}{3}, $ |
| Median | $ F^{-1}(0,5) = \begin{cases} a+\frac{\sqrt{2(b-a)(c-a)}}{2}, & \mbox{wenn } c\!\ge\!\frac{b\!-\!a}{2} \\ b-\frac{\sqrt{2(b-a)(b-c)}}{2}, & \mbox{wenn } c\!\le\!\frac{b\!-\!a}{2} \end{cases} $ |
| Varianz | $ \operatorname{var}(x) = \frac{a^2+b^2+c^2-ab-ac-bc}{18} $ |
| Standardabweichung | $ \sigma = \frac{1}{6} \sqrt{2(a^2+b^2+c^2-ab-ac-bc)} $ |
| Schiefe | $ \frac{\sqrt 2 (a\!+\!b\!-\!2c)(2a\!-\!b\!-\!c)(a\!-\!2b\!+\!c)}{5(a^2\!+\!b^2\!+\!c^2\!-\!ab\!-\!ac\!-\!bc)^\frac{3}{2}} $ |
| Wölbung | $ \frac{12}{5} $ |
| Entropie | $ \frac{1}{2}+\ln\left(\frac{b-a}{2}\right) $ |
| Momenterzeugende Funktion | $ 2\frac{(b\!-\!c)e^{at}\!-\!(b\!-\!a)e^{ct}\!+\!(c\!-\!a)e^{bt}} {(b-a)(c-a)(b-c)t^2} $ |
| Charakteristische Funktion | $ -2\frac{(b\!-\!c)e^{iat}\!-\!(b\!-\!a)e^{ict}\!+\!(c\!-\!a)e^{ibt}} {(b-a)(c-a)(b-c)t^2} $ |
