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math stat oef -- +ch3 oef4

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commit ffdd04535da84e6e961157ed91ebdc59483b2f84 1 parent 7412526
@ward authored
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  1. +90 −0 mathematicalstatistics/oefeningen.tex
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90 mathematicalstatistics/oefeningen.tex
@@ -26,6 +26,7 @@
\newcommand{\C}{{\mathbb C}}
\newcommand{\HQ}{{\mathbb H}}
+\newcommand{\Var}{\mathrm{Var}}
\newcommand{\Prob}{{\mathbb P}}
\newcommand{\E}{{\mathbb E}}
@@ -449,6 +450,95 @@ \section{Hypotheses testing}
\begin{oplossing}
\end{oplossing}
+\section{Linear Regression}
+
+\begin{opgave}
+\end{opgave}
+\begin{oplossing}
+\end{oplossing}
+
+\begin{opgave}
+\end{opgave}
+\begin{oplossing}
+\end{oplossing}
+
+\begin{opgave}
+\end{opgave}
+\begin{oplossing}
+\end{oplossing}
+
+\begin{opgave}%TODO Change the assignment \mu into \mu_0 to differentiate?
+ Two laboraties each take $c$ measurements on the same standard $\mu$.
+ Consider the model
+ \begin{align*}
+ Y_{ij} &= \mu + \epsilon_{ij} &&i = 1, 2 \text{ and } j = 1, \dots, c
+ \end{align*}
+ ... %TODO complete
+\end{opgave}
+\begin{oplossing}
+ \begin{align*}
+ \Var(Y_{1j}) &= \sigma^2\\
+ \Var(Y_{2j}) &= 4\sigma^2
+ \end{align*}
+
+ First make the variances equal.
+ \[
+ \begin{cases}
+ Y_{1j} = \mu + \epsilon_{1j}\\
+ \frac{Y_{2j}}{2} = \frac{\mu}{2} + \frac{\epsilon_{2j}}{2}
+ \end{cases}
+ \]
+
+ The Gaussian linear model.
+ \begin{align*}
+ \mathbf{Y} &= C\mathbf{\beta} + \mathbf{\epsilon}\\
+ &= \begin{pmatrix}1\\\vdots\\1\\0.5\\\vdots\\0.5\end{pmatrix}
+ \begin{pmatrix}\mu\end{pmatrix}
+ + \begin{pmatrix}
+ \epsilon_{11}\\\vdots\\\epsilon_{1c}\\
+ \epsilon_{21}\\\vdots\\\epsilon_{2c}
+ \end{pmatrix}
+ \end{align*}
+ with $C \in \R^{2c \times 1}$, $\mathbf{\beta} \in \R^{1 \times 1}$ and
+ $\mathbf{\epsilon} \in \R^{2c \times 1}$. The $\mu$ is the one from the
+ assignment, not the one that equals $C\mathbf{\beta}$ (afaik).
+
+ In other words to estimate the $\mu$ that they ask for, we need to estimate
+ $\beta$ in our Gaussian linear model.
+
+ \begin{align*}
+ \begin{pmatrix}\mu\end{pmatrix} &= \hat{\beta}\\
+ &= (C^{t} C)^{-1} C^{t} \mathbf{Y}\\
+ &= \left[
+ \begin{pmatrix}1 &\dots &1 &0.5 &\dots &0.5\end{pmatrix}
+ \begin{pmatrix}1\\\vdots\\1\\0.5\\\vdots\\0.5\end{pmatrix}
+ \right]^{-1}
+ \begin{pmatrix}1 &\dots &1 &0.5 &\dots &0.5\end{pmatrix}
+ \mathbf{Y}\\
+ &= \left[
+ \begin{pmatrix}c + \frac{c}{4}\end{pmatrix}
+ \right]^{-1}
+ \begin{pmatrix}1 &\dots &1 &0.5 &\dots &0.5\end{pmatrix}
+ \mathbf{Y}\\
+ &= \begin{pmatrix}\frac{4}{5c}\end{pmatrix}
+ \begin{pmatrix}1 &\dots &1 &0.5 &\dots &0.5\end{pmatrix}
+ \mathbf{Y}\\
+ &= \begin{pmatrix}\frac{4}{5c} &\dots &\frac{4}{5c} &\frac{2}{5c} &\dots &\frac{2}{5c}\end{pmatrix}
+ \begin{pmatrix}Y_{11}\\\vdots\\Y_{1c}\\Y_{21}/2\\\vdots\\Y_{2c}/2\end{pmatrix}
+ \end{align*}
+ and we get what we needed to prove.
+\end{oplossing}
+
+\begin{opgave}
+\end{opgave}
+\begin{oplossing}
+\end{oplossing}
+
+\begin{opgave}
+\end{opgave}
+\begin{oplossing}
+\end{oplossing}
+
%\newpage
%\begin{thebibliography}{99}
% \bibitem{VOORBEELD} Voorbeeld, Ward Muylaert.
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