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Expand file tree Collapse file tree Original file line number Diff line number Diff line change @@ -486,6 +486,7 @@ title: "Table of Contents"
486486 &emsp ;&ensp ; 3.10.3. ** [ Moment-generating function] ( /P/wald-mgf ) ** <br >
487487 &emsp ;&ensp ; 3.10.4. ** [ Mean] ( /P/wald-mean ) ** <br >
488488 &emsp ;&ensp ; 3.10.5. ** [ Variance] ( /P/wald-var ) ** <br >
489+ &emsp ;&ensp ; 3.10.6. ** [ Skewness] ( /P/wald-skew ) ** <br >
489490
490491 3.11. ex-Gaussian distribution <br >
491492 &emsp ;&ensp ; 3.11.1. * [ Definition] ( /D/exg ) * <br >
Original file line number Diff line number Diff line change @@ -70,15 +70,13 @@ $$ \label{eq:wald-moment}
7070\mathrm{E}(X^3) = M_X'''(0)
7171$$
7272
73- based on the [ relationship between raw moment and moment-generating function] ( /P/mom-mgf ) .
74-
75- First we differentiate
73+ based on the [ relationship between raw moment and moment-generating function] ( /P/mom-mgf ) . First, we differentiate the moment-generating function
7674
7775$$ \label{eq:wald-mgf}
7876M_X(t) = \exp\left[\alpha \gamma - \sqrt{\alpha^2(\gamma^2-2t)}\right]
7977$$
8078
81- with respect to $t$. Using the chain rule gives
79+ with respect to $t$. Using the chain rule, we have:
8280
8381$$ \label{eq:wald-skew-s1}
8482\begin{split}
@@ -118,7 +116,7 @@ $$ \label{eq:wald-skew-split2}
118116g(t) = \frac{\alpha}{(\gamma^2-2t)^{3/2}}\exp\left[\alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \; .
119117$$
120118
121- With this decomposition, $M '''(t) = f'(t) + g'(t)$. Applying the product rule to $f$ gives:
119+ With this decomposition, $M_X '''(t) = f'(t) + g'(t)$. Applying the product rule to $f$ gives:
122120
123121$$ \label{eq:wald-skew-f}
124122\begin{split}
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