Skip to content

Commit 87b7e77

Browse files
committed
edited index "ToC"
Proof "wald-skew" was added to index "ToC".
1 parent 3851834 commit 87b7e77

2 files changed

Lines changed: 4 additions & 5 deletions

File tree

I/ToC.md

Lines changed: 1 addition & 0 deletions
Original file line numberDiff line numberDiff 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>

P/wald-skew.md

Lines changed: 3 additions & 5 deletions
Original file line numberDiff line numberDiff 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
$$ \label{eq:wald-mgf}
7876
M_X(t) = \exp\left[\alpha \gamma - \sqrt{\alpha^2(\gamma^2-2t)}\right]
7977
$$
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}
118116
g(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}

0 commit comments

Comments
 (0)