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#LyX 1.6.2 created this file. For more info see http://www.lyx.org/
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\use_default_options true
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\font_typewriter default
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\font_default_family default
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\paperfontsize default
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\paperorientation portrait
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\paragraph_separation indent
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\quotes_language english
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\paperpagestyle default
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\tracking_changes false
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\begin_layout Standard
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\begin_layout Standard
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\begin_inset Formula $\boldsymbol{1}$
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\begin_layout Standard
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\begin_inset Formula $\boldsymbol{X}^{2}$
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\begin_layout Standard
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\begin_inset Formula $np(1-p)\text{ where }p=\frac{\mu}{n};\,\, V(\mu)=np(1-p)$
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\begin_layout Standard
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\begin_inset Formula $\frac{\partial\mu}{\partial\eta}$
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\begin_layout Standard
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Links: initialization of base class returns the actual mean vector
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\begin_inset Formula $\boldsymbol{\mu}$
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\begin_inset Formula $p$
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in the logit and subclasses;
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\begin_inset Formula $x$
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\begin_layout Standard
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\begin_inset Float table
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\begin_layout Plain Layout
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<lyxtabular version="3" rows="11" columns="4">
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<column alignment="center" valignment="top" width="0">
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<column alignment="center" valignment="top" width="0">
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<column alignment="center" valignment="top" width="0">
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<column alignment="center" valignment="top" width="0">
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $g(p)$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $g^{-1}(p)$
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $g^{\prime}(p)$
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $z=\log\frac{p}{1-p}$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $p=\frac{e^{z}}{1+e^{z}}$
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $g^{\prime}(p)=\frac{1}{p(1-p)}$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $z=x^{\text{pow}}$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $x=z^{\frac{1}{\text{pow}}}$
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $g^{\prime}(x)=\text{pow}\cdot x^{\text{power}-1}$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell multicolumn="1" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $\text{pow}=-1$
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<cell multicolumn="2" alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell multicolumn="2" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell multicolumn="1" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $\text{pow}=0.5$
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<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell multicolumn="1" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $\text{pow}=1$
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<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
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\begin_layout Plain Layout
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<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $z=\log x$
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<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $g^{-1}(z)=e^{z}$
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $g^{\prime}(x)=\frac{1}{x}$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $z=\Phi^{-1}(p)$
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<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $p=\Phi(z)$
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $g^{\prime}(x)=\frac{1}{\int_{-\infty}^{p}f(t)dt}$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell multicolumn="1" alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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same as the above with the Cauchy distribution
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<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
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\begin_layout Plain Layout
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<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $z=\log(-\log p)$
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<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $p=e^{-e^{z}}$
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $g^{\prime}(p)=-\frac{1}{p\log p}$
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\begin_layout Plain Layout
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\begin_layout Plain Layout
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\begin_layout Standard
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Initializing the family sets a link property and a variance based on the
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\begin_layout Standard
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\begin_inset Float table
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\begin_layout Plain Layout
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<lyxtabular version="3" rows="8" columns="6">
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<column alignment="center" valignment="top" width="0.75in">
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<column alignment="center" valignment="top" width="0">
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<column alignment="center" valignment="top" width="0">
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<column alignment="center" valignment="top" width="0">
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<column alignment="center" valignment="top" width="0">
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<column alignment="center" valignment="top" width="0">
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $\frac{1}{(g^{\prime}(\mu))^{2}\cdot V(\mu)}$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $\frac{\sum_{i}\text{DevResid}^{2}}{\text{scale}}$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $\left(Y-\mu\right)\cdot\sqrt{\text{weights}}$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $\mu=g^{-1}(\eta)$
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $\eta=g(\mu)$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $\text{sign}\left(Y-\mu\right)\sqrt{2Y\log\frac{Y}{\mu}-2(Y-\mu)}$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $\frac{\left(Y-\mu\right)}{\text{\sqrt{\text{scale}\cdot V\left(\mu\right)}}}$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_inset Formula $\text{sign}\left(Y-\mu\right)\sqrt{-2Y\log\frac{\mu}{n}+\left(n-Y\right)\log\left(1-\frac{\mu}{n}\right)}$
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_layout Plain Layout
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\begin_inset Caption
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\begin_layout Plain Layout
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\begin_layout Plain Layout
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\begin_inset Formula $\eta$
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is the linear predictor ie.,
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\begin_inset Formula $X\beta$
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in the generalized linear model