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}{ | 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>d := b-a\!</math><br><math>m := \frac{c-a}{d},\,1-m=\frac{b-c}{d},\,c = a+md = b - (1-m)d</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)}{ | \frac{2(x-a)}{d(c-a)} = \frac{2(x-a)}{md^2}, & \mbox{wenn } a \le x \le c \\ | ||
\frac{2(b-x)}{ | \frac{2(b-x)}{d(b-c)} = \frac{2(x-a)}{(1-m)d^2}, & \mbox{wenn } c < x \le b \\ | ||
0, & \mbox{sonst } | 0, & \mbox{sonst } | ||
\end{cases} | \end{cases} | ||
| Zeile 20: | Zeile 20: | ||
F(x) = | F(x) = | ||
\begin{cases} | \begin{cases} | ||
0, | 0, & \mbox{wenn } x < a\\ | ||
0+\frac{(x-a)^2}{ | 0+\frac{(x-a)^2}{d(c-a)} = 0+\frac{(x-a)^2}{md^2}, & \mbox{wenn } a \le x \le c \\ | ||
1-\frac{(b-x)^2}{ | 1-\frac{(b-x)^2}{d(b-c)} = 1-\frac{(b-x)^2}{(1-m)d^2}, & \mbox{wenn } c < x \le b \\ | ||
1, | 1, & \mbox{wenn } b < x | ||
\end{cases} | \end{cases} | ||
</math>| | </math>| | ||
mode =<math>c = a+ | mode =<math>c = a+md,\,f(c)=\frac{2}{d}\!</math>| | ||
mean =<math>\mu = \frac{a+b+c}{3} = a+\frac{(1+m) | mean =<math>\mu = \frac{a+b+c}{3} = a+\frac{(1+m)d}{3}</math>| | ||
quartile = <math> | quartile = <math> | ||
F^{-1}(p) = | F^{-1}(p) = | ||
\begin{cases} | \begin{cases} | ||
a+ | a+d\sqrt{mp}, & \mbox{wenn } 0 \le p \le m \\ | ||
b- | b-d\sqrt{(1-m)(1-p)}, & \mbox{wenn } m < p \le 1 | ||
\end{cases} | \end{cases} | ||
</math>| | </math>| | ||
| Zeile 38: | Zeile 38: | ||
F^{-1}(0,5) = | F^{-1}(0,5) = | ||
\begin{cases} | \begin{cases} | ||
a+\frac{\sqrt{ | a+\frac{\sqrt{2d(c-a)}}{2} = a+d\frac{\sqrt{2m}}{2}, & \mbox{wenn } 0{,}5 \le m \mbox{ bzw. } \frac{b+a}{2} \le c\\ | ||
b-\frac{\sqrt{ | b-\frac{\sqrt{2d(b-c)}}{2} = b-d\frac{\sqrt{2(1-m)}}{2}, & \mbox{wenn } m < 0{,}5 \mbox{ bzw. } c \le \frac{b+a}{2} | ||
\end{cases} | \end{cases} | ||
</math>| | </math>| | ||
variance =<math>\operatorname{var}(x) = \frac{a^2+b^2+c^2-ab-ac-bc}{18} = \frac{ | variance =<math>\operatorname{var}(x) = \frac{a^2+b^2+c^2-ab-ac-bc}{18} = \frac{d^2(1+m+m^2)}{18}</math>| | ||
sigma =<math>\sigma = \frac{1}{6} \sqrt{2(a^2+b^2+c^2-ab-ac-bc)} = \frac{1}{6} \sqrt{ | sigma =<math>\sigma = \frac{1}{6} \sqrt{2(a^2+b^2+c^2-ab-ac-bc)} = \frac{1}{6} \sqrt{2d^2(1+m+m^2)})</math>| | ||
skewness =<math> | skewness =<math> | ||
\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}} | \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}} | ||
</math>| | </math>| | ||
kurtosis =<math>\frac{12}{5}</math>| | kurtosis =<math>\frac{12}{5}</math>| | ||
entropy =<math>\frac{1}{2}+\ln\left(\frac{ | entropy =<math>\frac{1}{2}+\ln\left(\frac{d}{2}\right)</math>| | ||
mgf =<math>2\frac{(b-c)e^{at}-(b-a)e^{ct}\!+(c-a)e^{bt}}{(b-a)(c-a)(b-c)t^2}</math>| | mgf =<math>2\frac{(b-c)e^{at}-(b-a)e^{ct}\!+(c-a)e^{bt}}{(b-a)(c-a)(b-c)t^2}</math>| | ||
char =<math>-2\frac{(b-c)e^{iat}-(b-a)e^{ict}+(c-a)e^{ibt}}{(b-a)(c-a)(b-c)t^2}</math>| | char =<math>-2\frac{(b-c)e^{iat}-(b-a)e^{ict}+(c-a)e^{ibt}}{(b-a)(c-a)(b-c)t^2}</math>| | ||
Version vom 29. Mai 2006, 08:29 Uhr
Achtung: Die folgenden Formeln können noch Fehler enthalten.
| Parameter | $ a \in ]-\infty,\infty[ $ $ b \in ]-\infty,\infty[,\,b>a $ $ c \in ]a,b[ $ $ d := b-a\! $ $ m := \frac{c-a}{d},\,1-m=\frac{b-c}{d},\,c = a+md = b - (1-m)d $ |
| Dichtefunktion | $ f(x) = \begin{cases} \frac{2(x-a)}{d(c-a)} = \frac{2(x-a)}{md^2}, & \mbox{wenn } a \le x \le c \\ \frac{2(b-x)}{d(b-c)} = \frac{2(x-a)}{(1-m)d^2}, & \mbox{wenn } c < x \le b \\ 0, & \mbox{sonst } \end{cases} $ |
| Stetigkeit | $ \mbox{f(x) ist stetig auf }]\infty,\infty[\! $ |
| Träger | $ ]a,b[ \! $ |
| Verteilungsfunktion | $ F(x) = \begin{cases} 0, & \mbox{wenn } x < a\\ 0+\frac{(x-a)^2}{d(c-a)} = 0+\frac{(x-a)^2}{md^2}, & \mbox{wenn } a \le x \le c \\ 1-\frac{(b-x)^2}{d(b-c)} = 1-\frac{(b-x)^2}{(1-m)d^2}, & \mbox{wenn } c < x \le b \\ 1, & \mbox{wenn } b < x \end{cases} $ |
| Modus | $ c = a+md,\,f(c)=\frac{2}{d}\! $ |
| Erwartungswert | $ \mu = \frac{a+b+c}{3} = a+\frac{(1+m)d}{3} $ |
| Median | $ F^{-1}(0,5) = \begin{cases} a+\frac{\sqrt{2d(c-a)}}{2} = a+d\frac{\sqrt{2m}}{2}, & \mbox{wenn } 0{,}5 \le m \mbox{ bzw. } \frac{b+a}{2} \le c\\ b-\frac{\sqrt{2d(b-c)}}{2} = b-d\frac{\sqrt{2(1-m)}}{2}, & \mbox{wenn } m < 0{,}5 \mbox{ bzw. } c \le \frac{b+a}{2} \end{cases} $ |
| Varianz | $ \operatorname{var}(x) = \frac{a^2+b^2+c^2-ab-ac-bc}{18} = \frac{d^2(1+m+m^2)}{18} $ |
| Standardabweichung | $ \sigma = \frac{1}{6} \sqrt{2(a^2+b^2+c^2-ab-ac-bc)} = \frac{1}{6} \sqrt{2d^2(1+m+m^2)}) $ |
| 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{d}{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} $ |
