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App. Stats.: Funktion noch mal geändert

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henrydatei 2019-03-05 18:33:11 +00:00
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Erasmus/Applied statistics/Coursework 1.pdf
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Erasmus/Applied statistics/Coursework 1.tex
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@ -236,8 +236,12 @@ The probability density function $f(t)$ is
\begin{align}
f(t) = \frac{2t\cdot\frac{\exp(-t^2)}{100}}{100} = \frac{t\cdot\exp(-t^2)}{5000}\notag
\end{align}
\textcolor{red}{After the PDF was changed we have}
{\color{red}\begin{align}
f(t) = \frac{2t\cdot\exp\left(\frac{-t^2}{100} \right) }{100} \notag
\end{align}}
\begin{center}
\begin{tikzpicture}
\begin{tikzpicture}[scale=0.9]
\begin{axis}[
xmin=0, xmax=5, xlabel=$x$,
ymin=0, ymax=0.0001, ylabel=$y$,
@ -249,6 +253,19 @@ The probability density function $f(t)$ is
\addplot+[mark=none] {(x*exp(-x^2))/5000};
\end{axis}
\end{tikzpicture}
\begin{tikzpicture}[scale=0.9]
\begin{axis}[
xmin=0, xmax=20, xlabel=$x$,
ymin=0, ymax=0.2, ylabel=$y$,
samples=400,
axis y line=middle,
axis x line=middle,
restrict y to domain=0:1,
domain=0:20
]
\addplot+[mark=none, color=red] {(2*x*exp(-x^2/100))/100};
\end{axis}
\end{tikzpicture}
\end{center}
The cumulative distribution function $F(t)$ is then
\begin{align}
@ -256,8 +273,13 @@ The cumulative distribution function $F(t)$ is then
&= \int_0^t \frac{\xi\cdot\exp(-\xi^2)}{5000}\,\diff\xi\notag \\
&= \frac{\exp(-t^2)\Big(\exp(t^2)-1\Big)}{10000} \notag
\end{align}
{\color{red}\begin{align}
F(t) &= \int_0^t f(\xi)\,\diff\xi \notag \\
&= \int_0^t \frac{2\xi\cdot\exp\left(\frac{-\xi^2}{100} \right) }{100}\,\diff\xi \notag \\
&= 1-\exp\left(\frac{-t^2}{100} \right) \notag
\end{align}}
\begin{center}
\begin{tikzpicture}
\begin{tikzpicture}[scale=0.9]
\begin{axis}[
xmin=0, xmax=5, xlabel=$x$,
ymin=0, ymax=0.0001, ylabel=$y$,
@ -269,12 +291,29 @@ The cumulative distribution function $F(t)$ is then
\addplot+[mark=none] {(exp(-x^2)*(exp(x^2)-1))/10000};
\end{axis}
\end{tikzpicture}
\begin{tikzpicture}[scale=0.9]
\begin{axis}[
xmin=0, xmax=20, xlabel=$x$,
ymin=0, ymax=1, ylabel=$y$,
samples=400,
axis y line=middle,
axis x line=middle,
restrict y to domain=0:1,
domain=0:20
]
\addplot+[mark=none,color=red] {1-exp(-x^2/100)};
\end{axis}
\end{tikzpicture}
\end{center}
For the survival function we get
\begin{align}
R(t) &= 1 - F(t) \notag \\
&= \frac{\exp(-t^2)+9999}{10000} \notag
\end{align}
{\color{red}\begin{align}
R(t) &= 1-F(t) \notag \\
&= \exp\left(\frac{-t^2}{100} \right) \notag
\end{align}}
\begin{center}
\begin{tikzpicture}[scale=0.9]
\begin{axis}[
@ -306,15 +345,34 @@ For the survival function we get
\end{axis}
\end{tikzpicture}
\end{center}
To get the reliability of the component at $t=7$ we simply evaluate $R(7)$ which is 0.9999.
\begin{center}
\begin{tikzpicture}
\begin{axis}[
xmin=0, xmax=20, xlabel=$x$,
ymin=0, ymax=1, ylabel=$y$,
samples=400,
axis y line=middle,
axis x line=middle,
restrict y to domain=0:1,
domain=0:20
]
\addplot+[mark=none,color=red] {exp(-x^2/100)};
\end{axis}
\end{tikzpicture}
\end{center}
To get the reliability of the component at $t=7$ we simply evaluate $R(7)$ which is 0.9999 \textcolor{red}{(0.6126)}.
The hazard function is defined as
\begin{align}
h(t) &= \frac{f(t)}{1-F(t)} \notag \\
&= \frac{2t}{9999\cdot \exp(t^2)+1} \notag
\end{align}
{\color{red}\begin{align}
h(t) &= \frac{f(t)}{1-F(t)} \notag \\
&= \frac{t}{50} \notag
\end{align}}
\begin{center}
\begin{tikzpicture}
\begin{tikzpicture}[scale=0.9]
\begin{axis}[
xmin=0, xmax=5, xlabel=$x$,
ymin=0, ymax=0.0001, ylabel=$y$,
@ -326,6 +384,19 @@ The hazard function is defined as
\addplot+[mark=none] {(2*x)/(9999*exp(x^2)+1)};
\end{axis}
\end{tikzpicture}
\begin{tikzpicture}[scale=0.9]
\begin{axis}[
xmin=0, xmax=20, xlabel=$x$,
ymin=0, ymax=1, ylabel=$y$,
samples=400,
axis y line=middle,
axis x line=middle,
restrict y to domain=0:1,
domain=0:20
]
\addplot+[mark=none,color=red] {x/50};
\end{axis}
\end{tikzpicture}
\end{center}
The hazard function describes how an item ages where $t$ affects the risk of failure. It is the frequency with which the item fails, expressed in failures per unit of time.