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Display information for equation id:math.2273.66 on revision:2273
* Page found: Vorurteilsfreie Schätzung des statistischen Operators zu einem festen Zeitpunkt (eq math.2273.66)
(force rerendering)Occurrences on the following pages:
Hash: a7b596ebc37fde20a217f0e8ce510e17
TeX (original user input):
\begin{align}
& =\sum\limits_{m,n}^{{}}{\left[ \left\langle {{r}_{m}} | {{w}_{n}} \right\rangle \left\langle {{w}_{n}} | {{r}_{m}} \right\rangle {{r}_{m}}\ln {{r}_{m}}-{{r}_{m}}\left\langle {{r}_{m}} \right|\ln R\left| {{w}_{n}} \right\rangle \left\langle {{w}_{n}} | {{r}_{m}} \right\rangle \right]} \\
& =\sum\limits_{m,n}^{{}}{\left[ {{\left| \left\langle {{r}_{m}} | {{w}_{n}} \right\rangle \right|}^{2}}{{r}_{m}}\left( \ln {{r}_{m}}-\ln {{R}_{n}} \right) \right]} \\
& =\sum\limits_{m,n}^{{}}{\left[ {{\left| \left\langle {{r}_{m}} | {{w}_{n}} \right\rangle \right|}^{2}}{{r}_{m}}\left( -\ln \frac{{{R}_{n}}}{{{r}_{m}}} \right) \right]} \\
\end{align}
TeX (checked):
{\begin{aligned}&=\sum \limits _{m,n}^{}{\left[\left\langle {{r}_{m}}|{{w}_{n}}\right\rangle \left\langle {{w}_{n}}|{{r}_{m}}\right\rangle {{r}_{m}}\ln {{r}_{m}}-{{r}_{m}}\left\langle {{r}_{m}}\right|\ln R\left|{{w}_{n}}\right\rangle \left\langle {{w}_{n}}|{{r}_{m}}\right\rangle \right]}\\&=\sum \limits _{m,n}^{}{\left[{{\left|\left\langle {{r}_{m}}|{{w}_{n}}\right\rangle \right|}^{2}}{{r}_{m}}\left(\ln {{r}_{m}}-\ln {{R}_{n}}\right)\right]}\\&=\sum \limits _{m,n}^{}{\left[{{\left|\left\langle {{r}_{m}}|{{w}_{n}}\right\rangle \right|}^{2}}{{r}_{m}}\left(-\ln {\frac {{R}_{n}}{{r}_{m}}}\right)\right]}\\\end{aligned}}
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<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><mo>=</mo><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mo>,</mo><mi>n</mi></mrow></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mo>|</mo><msub><mi>w</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>w</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo>|</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mi>ln</mi><mo>⁡</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mo>−</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><mi>ln</mi><mo>⁡</mo><mi>R</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>w</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>w</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo>|</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mo>,</mo><mi>n</mi></mrow></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mo>|</mo><msub><mi>w</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ln</mi><mo>⁡</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mo>−</mo><mi>ln</mi><mo>⁡</mo><msub><mi>R</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>m</mi><mo>,</mo><mi>n</mi></mrow></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mo>|</mo><msub><mi>w</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mi>ln</mi><mo>⁡</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msub><mi>R</mi><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></msub></mrow><mrow data-mjx-texclass="ORD"><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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