Merge pull request #116 from StatProofBook/master

JoramSoch · web-flow · commit 636a2af70101 · 2023-10-27T12:35:13.000+02:00
update to master
diff --git a/D/anova1.md b/D/anova1.md
@@ -33,7 +33,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Consider measurements $y_{ij} \in \mathbb{R}$ from disctinct objects $j = 1, \ldots, n_i$ in separate groups $i = 1, \ldots, k$.
+**Definition:** Consider measurements $y_{ij} \in \mathbb{R}$ from distinct objects $j = 1, \ldots, n_i$ in separate groups $i = 1, \ldots, k$.
 
 Then, in one-way analysis of variance (ANOVA), these measurements are assumed to come from [normal distributions](/D/norm)
 
diff --git a/D/bin-data.md b/D/bin-data.md
@@ -21,4 +21,4 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** An ordered pair $(n,y)$ with $n \in \mathbb{N}$ and $y \in \mathbb{N}_0$, where $y$ is the number of successes in $n$ trials, consititutes a set of binomial observations.
+**Definition:** An ordered pair $(n,y)$ with $n \in \mathbb{N}$ and $y \in \mathbb{N}_0$, where $y$ is the number of successes in $n$ trials, constitutes a set of binomial observations.
diff --git a/D/chi2.md b/D/chi2.md
@@ -45,7 +45,7 @@ $$\label{eq:chi2}
 Y = \sum_{i=1}^{k} X_{i}^{2} \sim \chi^{2}(k) \quad \text{where} \quad k > 0 \; .
 $$
 
-The [probability density function of the chi-squared distribution](/P/chi2-pdf) with $k$ degress of freedom is
+The [probability density function of the chi-squared distribution](/P/chi2-pdf) with $k$ degrees of freedom is
 
 $$ \label{eq:chi2-pdf}
 \chi^{2}(x; k) = \frac{1}{2^{k/2}\Gamma (k/2)} \, x^{k/2-1} \, e^{-x/2}
diff --git a/D/mult-data.md b/D/mult-data.md
@@ -21,4 +21,4 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** An ordered pair $(n,y)$ with $n \in \mathbb{N}$ and $y = \left[ y_1, \ldots, y_k \right] \in \mathbb{N}_0^{1 \times k}$, where $y_i$ is the number of observations for the $i$-th out of $k$ categories obtained in $n$ trials, $i = 1, \ldots, k$, consititutes a set of multinomial observations.
+**Definition:** An ordered pair $(n,y)$ with $n \in \mathbb{N}$ and $y = \left[ y_1, \ldots, y_k \right] \in \mathbb{N}_0^{1 \times k}$, where $y_i$ is the number of observations for the $i$-th out of $k$ categories obtained in $n$ trials, $i = 1, \ldots, k$, constitutes a set of multinomial observations.
diff --git a/I/PbA.md b/I/PbA.md
@@ -8,7 +8,7 @@ title: "Proof by Author"
 
 - [Covariance matrix of the multinomial distribution](/P/mult-cov)
 
-### JoramSoch (391 proofs)
+### JoramSoch (392 proofs)
 
 - [Accuracy and complexity for the univariate Gaussian](/P/ug-anc)
 - [Accuracy and complexity for the univariate Gaussian with known variance](/P/ugkv-anc)
@@ -139,6 +139,7 @@ title: "Proof by Author"
 - [Inverse transformation method using cumulative distribution function](/P/cdf-itm)
 - [Joint likelihood is the product of likelihood function and prior density](/P/jl-lfnprior)
 - [Kullback-Leibler divergence for the Bernoulli distribution](/P/bern-kl)
+- [Kullback-Leibler divergence for the binomial distribution](/P/bin-kl)
 - [Kullback-Leibler divergence for the Dirichlet distribution](/P/dir-kl)
 - [Kullback-Leibler divergence for the gamma distribution](/P/gam-kl)
 - [Kullback-Leibler divergence for the matrix-normal distribution](/P/matn-kl)
@@ -426,7 +427,7 @@ title: "Proof by Author"
 
 - [Proof Template](/P/-temp-)
 
-### tomfaulkenberry (15 proofs)
+### tomfaulkenberry (16 proofs)
 
 - [Encompassing prior method for computing Bayes factors](/P/bf-ep)
 - [Mean of the ex-Gaussian distribution](/P/exg-mean)
@@ -440,6 +441,7 @@ title: "Proof by Author"
 - [Savage-Dickey density ratio for computing Bayes factors](/P/bf-sddr)
 - [Skewness of the ex-Gaussian distribution](/P/exg-skew)
 - [Skewness of the exponential distribution](/P/exp-skew)
+- [Skewness of the Wald distribution](/P/wald-skew)
 - [Transitivity of Bayes Factors](/P/bf-trans)
 - [Variance of the ex-Gaussian distribution](/P/exg-var)
 - [Variance of the Wald distribution](/P/wald-var)
diff --git a/I/PbN.md b/I/PbN.md
@@ -425,3 +425,5 @@ title: "Proof by Number"
 | P417 | bvn-pdfcorr | [Probability density function of the bivariate normal distribution in terms of correlation coefficient](/P/bvn-pdfcorr) | JoramSoch | 2023-09-29 |
 | P418 | mlr-olstr | [Ordinary least squares for multiple linear regression with two regressors](/P/mlr-olstr) | JoramSoch | 2023-10-06 |
 | P419 | bern-kl | [Kullback-Leibler divergence for the Bernoulli distribution](/P/bern-kl) | JoramSoch | 2023-10-13 |
+| P420 | bin-kl | [Kullback-Leibler divergence for the binomial distribution](/P/bin-kl) | JoramSoch | 2023-10-20 |
+| P421 | wald-skew | [Skewness of the Wald distribution](/P/wald-skew) | tomfaulkenberry | 2023-10-24 |
diff --git a/I/PbT.md b/I/PbT.md
@@ -166,6 +166,7 @@ title: "Proof by Topic"
 ### K
 
 - [Kullback-Leibler divergence for the Bernoulli distribution](/P/bern-kl)
+- [Kullback-Leibler divergence for the binomial distribution](/P/bin-kl)
 - [Kullback-Leibler divergence for the Dirichlet distribution](/P/dir-kl)
 - [Kullback-Leibler divergence for the gamma distribution](/P/gam-kl)
 - [Kullback-Leibler divergence for the matrix-normal distribution](/P/matn-kl)
@@ -438,6 +439,7 @@ title: "Proof by Topic"
 - [Simple linear regression is a special case of multiple linear regression](/P/slr-mlr)
 - [Skewness of the ex-Gaussian distribution](/P/exg-skew)
 - [Skewness of the exponential distribution](/P/exp-skew)
+- [Skewness of the Wald distribution](/P/wald-skew)
 - [Square of expectation of product is less than or equal to product of expectation of squares](/P/mean-prodsqr)
 - [Sums of squares for simple linear regression](/P/slr-sss)
 - [Symmetry of the covariance](/P/cov-symm)
diff --git a/I/PwS.md b/I/PwS.md
@@ -136,6 +136,7 @@ title: "Proofs without Source"
 - [Simple linear regression is a special case of multiple linear regression](/P/slr-mlr)
 - [Skewness of the ex-Gaussian distribution](/P/exg-skew)
 - [Skewness of the exponential distribution](/P/exp-skew)
+- [Skewness of the Wald distribution](/P/wald-skew)
 - [Sums of squares for simple linear regression](/P/slr-sss)
 - [Transformation matrices for simple linear regression](/P/slr-mat)
 - [Transitivity of Bayes Factors](/P/bf-trans)
diff --git a/I/ToC.md b/I/ToC.md
@@ -486,6 +486,7 @@ title: "Table of Contents"
    &emsp;&ensp; 3.10.3. **[Moment-generating function](/P/wald-mgf)** <br>
    &emsp;&ensp; 3.10.4. **[Mean](/P/wald-mean)** <br>
    &emsp;&ensp; 3.10.5. **[Variance](/P/wald-var)** <br>
+   &emsp;&ensp; 3.10.6. **[Skewness](/P/wald-skew)** <br>
    
    3.11. ex-Gaussian distribution <br>
    &emsp;&ensp; 3.11.1. *[Definition](/D/exg)* <br>
diff --git a/P/bf-trans.md b/P/bf-trans.md
@@ -21,7 +21,7 @@ username: "tomfaulkenberry"
 ---
 
 
-**Theorem:** Consider three competing [models](/D/gm) $m_1$, $m_2$, and $m3$ for observed data $y$. Then the [Bayes factor](/D/bf) for $m_1$ over $m_3$ can be written as:
+**Theorem:** Consider three competing [models](/D/gm) $m_1$, $m_2$, and $m_33$ for observed data $y$. Then the [Bayes factor](/D/bf) for $m_1$ over $m_3$ can be written as:
 
 $$ \label{eq:bf-trans}
 \text{BF}_{13} = \text{BF}_{12}\cdot \text{BF}_{23}.
diff --git a/P/bin-kl.md b/P/bin-kl.md
@@ -54,15 +54,15 @@ which, applied to the [binomial distributions](/D/bin) in \eqref{eq:bins}, yield
 $$ \label{eq:bin-KL-s1}
 \begin{split}
 \mathrm{KL}[P\,||\,Q] &= \sum_{x=0}^{n} p(x) \, \ln \frac{p(x)}{q(x)} \\
-&= p(X=0) \cdot \ln \frac{p(X=0)}{q(X=0)} + \ldots + p(X=0) \cdot \ln \frac{p(X=0)}{q(X=0)} \; .
+&= p(X=0) \cdot \ln \frac{p(X=0)}{q(X=0)} + \ldots + p(X=n) \cdot \ln \frac{p(X=n)}{q(X=n)} \; .
 \end{split}
 $$
 
 Using the [probability mass function of the binomial distribution](/P/bin-pmf), this becomes:
 
 $$ \label{eq:bin-KL-s2}
 \begin{split}
-\mathrm{KL}[P\,||\,Q] &= \sum_{x=0}^{n} {n \choose x} \, p_1^x \, (1-p_1)^{n-x} \cdot \ln \frac{{n \choose x} \, p_1^x \, (1-p_1)^{n-x}}{{n \choose x} \, p_2^x \, (1-p_2)^{n-x}} \\
+\mathrm{KL}[P\,||\,Q] &= \sum_{x=0}^{n} {n \choose x} \, p_1^x \, (1-p_1)^{n-x} \cdot \ln \frac{ {n \choose x} \, p_1^x \, (1-p_1)^{n-x} }{ {n \choose x} \, p_2^x \, (1-p_2)^{n-x} } \\
 &= \sum_{x=0}^{n} {n \choose x} \, p_1^x \, (1-p_1)^{n-x} \cdot \left[ x \cdot \ln \frac{p_1}{p_2} + (n-x) \cdot \ln \frac{1-p_1}{1-p_2} \right] \\
 &= \ln \frac{p_1}{p_2} \cdot \sum_{x=0}^{n} {n \choose x} \, p_1^x \, (1-p_1)^{n-x} x + \ln \frac{1-p_1}{1-p_2} \cdot \sum_{x=0}^{n} {n \choose x} \, p_1^x \, (1-p_1)^{n-x} (n-x) \; .
 \end{split}
diff --git a/P/bvn-pdf.md b/P/bvn-pdf.md
@@ -21,7 +21,7 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let $X = \left[ \begin{matrix} X_1 & X_2 \end{matrix} \right]^\mathrm{T}$ follow a [bivariate normal distribution](/D/bvn):
+**Theorem:** Let $X = \left[ \begin{matrix} X_1 \\\\ X_2 \end{matrix} \right]$ follow a [bivariate normal distribution](/D/bvn):
 
 $$ \label{eq:bvn}
 X \sim \mathcal{N}\left( \mu = \left[ \begin{matrix} \mu_1 \\ \mu_2 \end{matrix} \right], \Sigma = \left[ \begin{matrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{matrix} \right] \right) \; .
diff --git a/P/chi2-gam.md b/P/chi2-gam.md
@@ -28,7 +28,7 @@ X \sim \mathrm{Gam}\left( \frac{k}{2}, \frac{1}{2} \right) \Rightarrow X \sim \c
 $$
 
 
-**Proof:** The [probability density function of the gamma distribution](/P/gam-pdf) for $x > 0$, where $\alpha$ is the shape parameter and $\beta$ is the rate paramete, is as follows:
+**Proof:** The [probability density function of the gamma distribution](/P/gam-pdf) for $x > 0$, where $\alpha$ is the shape parameter and $\beta$ is the rate parameter, is as follows:
 
 $$ \label{eq:gam-pdf}
 \mathrm{Gam}(x; \alpha, \beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \, x^{\alpha-1} \, e^{-\beta x}
diff --git a/P/mlr-wls2.md b/P/mlr-wls2.md
@@ -80,7 +80,7 @@ $$ \label{eq:wRSS-der}
 \frac{\mathrm{d}\mathrm{wRSS}(\beta)}{\mathrm{d}\beta} = - 2 X^\mathrm{T} V^{-1} y + 2 X^\mathrm{T} V^{-1} X \beta
 $$
 
-and setting this deriative to zero, we obtain:
+and setting this derivative to zero, we obtain:
 
 $$ \label{eq:WLS-qed}
 \begin{split}
diff --git a/P/prob-exh2.md b/P/prob-exh2.md
@@ -40,7 +40,7 @@ $$ \label{eq:prob-exh}
 $$
 
 
-**Proof:** The [addition law of probability](/P/prob-add) states that for two [events](/D/reve) $A$ and $B$, the [probability](/D/prob) of at least one of them occuring is:
+**Proof:** The [addition law of probability](/P/prob-add) states that for two [events](/D/reve) $A$ and $B$, the [probability](/D/prob) of at least one of them occurring is:
 
 $$ \label{eq:prob-add}
 P(A \cup B) = P(A) + P(B) - P(A \cap B) \; .
diff --git a/P/slr-ols2.md b/P/slr-ols2.md
@@ -45,7 +45,7 @@ $$ \label{eq:slr-mlr}
 X = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right]
 $$
 
-and [ordinary least sqaures estimates](/P/mlr-ols) are given by
+and [ordinary least squares estimates](/P/mlr-ols) are given by
 
 $$ \label{eq:mlr-ols}
 \hat{\beta} = (X^\mathrm{T} X)^{-1} X^\mathrm{T} y \; .
diff --git a/P/slr-sss.md b/P/slr-sss.md
@@ -64,7 +64,7 @@ $$ \label{eq:ESS}
 \mathrm{ESS} = \sum_{i=1}^n (\hat{y}_i - \bar{y})^2 \quad \text{where} \quad \hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_i
 $$
 
-which, with the OLS parameter estimats, becomes:
+which, with the OLS parameter estimates, becomes:
 
 $$ \label{eq:ESS-qed}
 \begin{split}
@@ -86,7 +86,7 @@ $$ \label{eq:RSS}
 \mathrm{RSS} = \sum_{i=1}^n (y_i - \hat{y}_i)^2 \quad \text{where} \quad \hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_i
 $$
 
-which, with the OLS parameter estimats, becomes:
+which, with the OLS parameter estimates, becomes:
 
 $$ \label{eq:RSS-qed}
 \begin{split}
diff --git a/P/slr-wls2.md b/P/slr-wls2.md
@@ -45,7 +45,7 @@ $$ \label{eq:slr-mlr}
 X = \left[ \begin{matrix} 1_n & x \end{matrix} \right] \quad \text{and} \quad \beta = \left[ \begin{matrix} \beta_0 \\ \beta_1 \end{matrix} \right]
 $$
 
-and [weighted least sqaures estimates](/P/mlr-wls) are given by
+and [weighted least squares estimates](/P/mlr-wls) are given by
 
 $$ \label{eq:mlr-wls}
 \hat{\beta} = (X^\mathrm{T} V^{-1} X)^{-1} X^\mathrm{T} V^{-1} y \; .
diff --git a/P/ug-prior.md b/P/ug-prior.md
@@ -67,7 +67,7 @@ $$
 using the inverse variance or precision $\tau = 1/\sigma^2$.
 
 <br>
-Seperating constant and variable terms, we have:
+Separating constant and variable terms, we have:
 
 $$ \label{eq:UG-LF-s1}
 p(y|\mu,\tau) = \sqrt{\frac{1}{(2 \pi)^n}} \cdot \tau^{n/2} \cdot \exp\left[ -\frac{\tau}{2} \sum_{i=1}^{n} \left( y_i-\mu \right)^2 \right] \; .
diff --git a/P/var-nonneg.md b/P/var-nonneg.md
@@ -41,7 +41,7 @@ $$ \label{eq:var}
 $$
 
 <br>
-1) If $X$ is a discrete [random variable](/D/rvar), then, because squares and probabilities are stricly non-negative, all the addends in
+1) If $X$ is a discrete [random variable](/D/rvar), then, because squares and probabilities are strictly non-negative, all the addends in
 
 $$ \label{eq:var-disc}
 \mathrm{Var}(X) = \sum_{x \in \mathcal{X}} (x-\mathrm{E}(X))^2 \cdot f_X(x)
diff --git a/P/wald-skew.md b/P/wald-skew.md
@@ -0,0 +1,164 @@
+---
+layout: proof
+mathjax: true
+
+author: "Thomas J. Faulkenberry"
+affiliation: "Tarleton State University"
+e_mail: "faulkenberry@tarleton.edu"
+date: 2023-10-24 12:00:00
+
+title: "Skewness of the Wald distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Wald distribution"
+theorem: "Skewness"
+
+sources:
+  
+proof_id: "P421"
+shortcut: "wald-skew"
+username: "tomfaulkenberry"
+---
+  
+
+
+**Theorem:** Let $X$ be a [random variable](/D/rvar) following a [Wald distribution](/D/wald):
+
+$$ \label{eq:wald}
+X \sim \mathrm{Wald}(\gamma,\alpha) \; .
+$$
+
+Then the [skewness](/D/skew) of $X$ is
+
+$$ \label{eq:wald-skew}
+\mathrm{Skew}(X) = \frac{3}{\sqrt{\alpha\gamma}} \; .
+$$
+
+**Proof:** 
+
+To compute the skewness of $X$, we [partition the skewness into expected values](/P/skew-mean):
+
+$$ \label{eq:skew-mean}
+\mathrm{Skew}(X) = \frac{\mathrm{E}(X^3)-3\mu\sigma^2-\mu^3}{\sigma^3} \; ,
+$$
+
+where $\mu$ and $\sigma$ are the mean and standard deviation of $X$, respectively. Since $X$ follows an [Wald distribution](/D/wald), the [mean](/P/wald-mean) of $X$ is given by 
+
+$$ \label{eq:wald-mean}
+\mu = \mathrm{E}(X) = \frac{\alpha}{\gamma}
+$$
+
+and the [standard deviation](/P/wald-var) of $X$ is given by
+
+$$ \label{eq:wald-var}
+\sigma = \sqrt{\mathrm{Var}(X)} = \sqrt{\frac{\alpha}{\gamma^3}}\; .
+$$
+
+Substituting \eqref{eq:wald-mean} and \eqref{eq:wald-var} into \eqref{eq:skew-mean} gives:
+
+$$ \label{eq:skew-mean-alt}
+\begin{split}
+\mathrm{Skew}(X) &= \frac{\mathrm{E}(X^3)-3\mu\sigma^2-\mu^3}{\sigma^3}\\
+&= \frac{\mathrm{E}(X^3) - 3\left(\frac{\alpha}{\gamma}\right)\left(\frac{\alpha}{\gamma^3}\right)-\left(\frac{\alpha}{\gamma}\right)^3}{\left(\sqrt{\frac{\alpha}{\gamma^3}}\right)^3}\\
+&= \frac{\gamma^{9/2}}{\alpha^{3/2}}\left[\mathrm{E}(X^3) - \frac{3\alpha^2}{\gamma^4}-\frac{\alpha^3}{\gamma^3}\right] \; .
+\end{split}
+$$
+
+Thus, the remaining work is to compute $\mathrm{E}(X^3)$. To do this, we use the [moment-generating function of the Wald distribution](/P/wald-mgf) to calculate
+
+$$ \label{eq:wald-moment}
+\mathrm{E}(X^3) = M_X'''(0)
+$$
+
+based on the [relationship between raw moment and moment-generating function](/P/mom-mgf). First, we differentiate the moment-generating function
+
+$$ \label{eq:wald-mgf}
+M_X(t) = \exp\left[\alpha \gamma - \sqrt{\alpha^2(\gamma^2-2t)}\right]
+$$
+
+with respect to $t$. Using the chain rule, we have:
+
+$$ \label{eq:wald-skew-s1}
+\begin{split}
+  M_X'(t) &= \exp\left[\alpha \gamma - \sqrt{\alpha^2(\gamma^2-2t)}\right] \cdot -\frac{1}{2}\left(\alpha^2(\gamma^2-2t)\right)^{-1/2}\cdot -2\alpha^2 \\
+          &= \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \cdot \frac{\alpha^2}{\sqrt{\alpha^2(\gamma^2-2t)}} \\
+          &= \alpha \cdot \exp\left[\alpha \gamma -\sqrt{\alpha^2(\gamma^2-2t)}\right] \cdot (\gamma^2-2t)^{-1/2} \; .
+\end{split}
+$$
+
+Now we use the product rule to obtain the second derivative:
+
+$$ \label{eq:wald-skew-s2}
+\begin{split}
+  M_X''(t) &= \alpha \cdot \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\cdot (\gamma^2-2t)^{-1/2}\cdot -\frac{1}{2}\left(\alpha^2(\gamma^2-2t)\right)^{-1/2}\cdot -2\alpha^2 \\
+           &+ \alpha \cdot \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\cdot -\frac{1}{2}(\gamma^2-2t)^{-3/2}\cdot -2 \\
+           &= \alpha^2\cdot \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\cdot (\gamma^2-2t)^{-1} \\
+           &+ \alpha\cdot \exp\left[\alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\cdot (\gamma^2-2t)^{-3/2} \\
+           & = \frac{\alpha^2}{\gamma^2-2t}\exp\left[\alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] + \frac{\alpha}{(\gamma^2-2t)^{3/2}}\exp\left[\alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \; .
+\end{split}
+$$
+
+Finally, one more application of the chain rule will give us the third derivative. To start, we will decompose the second derivative obtained in \eqref{eq:wald-skew-s2} as
+
+$$ \label{eq:wald-skew-s3}
+M''(t) = f(t) + g(t)
+$$
+
+where
+
+$$ \label{eq:wald-skew-split1}
+f(t) = \frac{\alpha^2}{\gamma^2-2t}\exp\left[\alpha\gamma-\sqrt{\alpha^2(\gamma^2-t2)}\right]
+$$
+
+and
+
+$$ \label{eq:wald-skew-split2}
+g(t) = \frac{\alpha}{(\gamma^2-2t)^{3/2}}\exp\left[\alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \; .
+$$
+
+With this decomposition, $M_X'''(t) = f'(t) + g'(t)$. Applying the product rule to $f$ gives:
+
+$$ \label{eq:wald-skew-f}
+\begin{split}
+  f'(t) &= 2\alpha^2(\gamma^2-2t)^{-2}\exp\left[\alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \\
+  &+ \alpha^2(\gamma^2-2t)^{-1}\exp\left[\alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\cdot -\frac{1}{2}\left[\alpha^2(\gamma^2-2t)\right]^{-1/2}\cdot -2\alpha^2\\
+  &= \frac{2\alpha^2}{(\gamma^2-2t)^2}\exp\left[\alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \\
+  &+ \frac{\alpha^3}{(\gamma^2-2t)^{3/2}}\exp\left[\alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \; .
+\end{split}
+$$
+
+Similarly, applying the product rule to $g$ gives:
+
+$$ \label{eq:wald-skew-g}
+\begin{split}
+  g'(t) &= -\frac{3}{2}\alpha(\gamma^2-2t)^{-5/2}(-2)\exp\left[\alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \\
+  &+ \alpha(\gamma^2-2t)^{-3/2}\exp\left[\alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\cdot -\frac{1}{2}\left[\alpha^2(\gamma^2-2t)\right]^{-1/2}\cdot -2\alpha^2\\
+&= \frac{3\alpha}{(\gamma^2-2t)^{5/2}}\exp\left[\alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \\
+&+ \frac{\alpha^2}{(\gamma^2-2t)^2}\exp\left[\alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \; .
+\end{split}\\
+$$
+
+Applying \eqref{eq:wald-moment}, together with \eqref{eq:wald-skew-f} and \eqref{eq:wald-skew-g}, yields
+
+$$ \label{eq:wald-skew-s4}
+\begin{split}
+\mathrm{E}(X^3) &= M_X'''(0)\\
+ &= f'(0) + g'(0)\\
+ &= \left[\frac{2\alpha^2}{\gamma^4}+\frac{\alpha^3}{\gamma^3}\right] + \left[\frac{3\alpha}{\gamma^5}+\frac{\alpha^2}{\gamma^4}\right]\\
+ &= \frac{3\alpha^2}{\gamma^4} + \frac{\alpha^3}{\gamma^3} + \frac{3\alpha}{\gamma^5} \; .
+\end{split}
+$$
+
+We now substitute \eqref{eq:wald-skew-s4} into \eqref{eq:skew-mean-alt}, giving
+
+$$ \label{eq:wald-skew-s5}
+\begin{split}
+\mathrm{Skew}(X) &= \frac{\gamma^{9/2}}{\alpha^{3/2}}\left[\mathrm{E}(X^3) - \frac{3\alpha^2}{\gamma^4}-\frac{\alpha^3}{\gamma^3}\right] \\
+&= \frac{\gamma^{9/2}}{\alpha^{3/2}}\left[\frac{3\alpha^2}{\gamma^4} + \frac{\alpha^3}{\gamma^3} + \frac{3\alpha}{\gamma^5} - \frac{3\alpha^2}{\gamma^4}-\frac{\alpha^3}{\gamma^3}\right] \\
+&= \frac{\gamma^{9/2}}{\alpha^{3/2}} \cdot \frac{3\alpha}{\gamma^5}\\
+&= \frac{3}{\alpha^{1/2}\cdot \gamma^{1/2}}\\
+&= \frac{3}{\sqrt{\alpha\gamma}} \; .
+\end{split}
+$$
+
+This completes the proof of \eqref{eq:wald-skew}.

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`@@ -21,4 +21,4 @@ username: "JoramSoch"`
`21`	`21`	`---`
`22`	`22`
`23`	`23`
`24`		`-Definition: An ordered pair $(n,y)$ with $n \in \mathbb{N}$ and $y \in \mathbb{N}_0$, where $y$ is the number of successes in $n$ trials, consititutes a set of binomial observations.`
	`24`	`+Definition: An ordered pair $(n,y)$ with $n \in \mathbb{N}$ and $y \in \mathbb{N}_0$, where $y$ is the number of successes in $n$ trials, constitutes a set of binomial observations.`