integrated new content

Proofs "wald-mgf/mean/var" were added to the table of contents; index pages updated.

Author: JoramSoch

I/Proof_by_Author.md

@@ -174,9 +174,12 @@ title: "Proof by Author"
 
 - [Proof Template](/P/-temp-)
 
-### tomfaulkenberry (4 proofs)
+### tomfaulkenberry (7 proofs)
 
 - [Encompassing Prior Method for computing Bayes Factors](/P/bf-ep)
+- [Mean of the Wald distribution](/P/wald-mean)
+- [Moment-generating function of the Wald distribution](/P/wald-mgf)
 - [Probability density function of the Wald distribution](/P/wald-pdf)
 - [Savage-Dickey Density Ratio for computing Bayes Factors](/P/bf-sddr)
 - [Transitivity of Bayes Factors](/P/bf-trans)
+- [Variance of the Wald distribution](/P/wald-var)

I/Proof_by_Number.md

@@ -173,4 +173,7 @@ title: "Proof by Number"
 | P164 | gibbs-ineq | [Gibbs' inequality](/P/gibbs-ineq) | JoramSoch | 2020-09-09 |
 | P165 | logsum-ineq | [Log sum inequality](/P/logsum-ineq) | JoramSoch | 2020-09-09 |
 | P166 | kl-nonneg2 | [Non-negativity of the Kullback-Leibler divergence](/P/kl-nonneg2) | JoramSoch | 2020-09-09 |
-| P167 | momcent-1st | [First central moment is zero](/P/momcent-1st) | JoramSoch | 2020-08-19 |
+| P167 | momcent-1st | [First central moment is zero](/P/momcent-1st) | JoramSoch | 2020-09-09 |
+| P168 | wald-mgf | [Moment-generating function of the Wald distribution](/P/wald-mgf) | tomfaulkenberry | 2020-09-13 |
+| P169 | wald-mean | [Mean of the Wald distribution](/P/wald-mean) | tomfaulkenberry | 2020-09-13 |
+| P170 | wald-var | [Variance of the Wald distribution](/P/wald-var) | tomfaulkenberry | 2020-09-13 |

I/Proof_by_Topic.md

@@ -114,6 +114,7 @@ title: "Proof by Topic"
 - [Maximum likelihood estimator of variance is biased](/P/resvar-bias)
 - [Mean of the Bernoulli distribution](/P/bern-mean)
 - [Mean of the Poisson distribution](/P/poiss-mean)
+- [Mean of the Wald distribution](/P/wald-mean)
 - [Mean of the binomial distribution](/P/bin-mean)
 - [Mean of the categorical distribution](/P/cat-mean)
 - [Mean of the continuous uniform distribution](/P/cuni-mean)
@@ -130,6 +131,7 @@ title: "Proof by Topic"
 - [Mode of the normal distribution](/P/norm-mode)
 - [Moment in terms of moment-generating function](/P/mom-mgf)
 - [Moment-generating function of linear combination of independent random variables](/P/mgf-lincomb)
+- [Moment-generating function of the Wald distribution](/P/wald-mgf)
 - [Moment-generating function of the normal distribution](/P/norm-mgf)
 - [Monotonicity of the expected value](/P/mean-mono)
 
@@ -226,6 +228,7 @@ title: "Proof by Topic"
 ### V
 
 - [Variance of constant is zero](/P/var-const)
+- [Variance of the Wald distribution](/P/wald-var)
 - [Variance of the gamma distribution](/P/gam-var)
 - [Variance of the linear combination of two random variables](/P/var-lincomb)
 - [Variance of the normal distribution](/P/norm-var)

I/Proofs_without_Source.md

@@ -25,6 +25,7 @@ title: "Proofs without Source"
 - [Maximum likelihood estimation for Poisson-distributed data](/P/poiss-mle)
 - [Maximum likelihood estimation for multiple linear regression](/P/mlr-mle)
 - [Maximum likelihood estimation for the general linear model](/P/glm-mle)
+- [Mean of the Wald distribution](/P/wald-mean)
 - [Mean of the categorical distribution](/P/cat-mean)
 - [Mean of the continuous uniform distribution](/P/cuni-mean)
 - [Mean of the multinomial distribution](/P/mult-mean)
@@ -62,5 +63,6 @@ title: "Proofs without Source"
 - [Relationship between signal-to-noise ratio and R²](/P/snr-rsq)
 - [Transitivity of Bayes Factors](/P/bf-trans)
 - [Transposition of a matrix-normal random variable](/P/matn-trans)
+- [Variance of the Wald distribution](/P/wald-var)
 - [Weighted least squares for multiple linear regression](/P/mlr-wls2)
 - [Weighted least squares for the general linear model](/P/glm-wls)

I/Table_of_Contents.md

@@ -226,16 +226,16 @@ title: "Table of Contents"
    &emsp;&ensp; 3.2.2. *[Standard normal distribution](/D/snorm)* <br>
    &emsp;&ensp; 3.2.3. **[Relation to standard normal distribution](/P/norm-snorm)** <br>
    &emsp;&ensp; 3.2.4. **[Probability density function](/P/norm-pdf)** <br>
-   &emsp;&ensp; 3.2.5. **[Cumulative distribution function](/P/norm-cdf)** <br>
-   &emsp;&ensp; 3.2.6. **[Cumulative distribution function without error function](/P/norm-cdfwerf)** <br>
-   &emsp;&ensp; 3.2.7. **[Quantile function](/P/norm-qf)** <br>
-   &emsp;&ensp; 3.2.8. **[Mean](/P/norm-mean)** <br>
-   &emsp;&ensp; 3.2.9. **[Median](/P/norm-med)** <br>
-   &emsp;&ensp; 3.2.10. **[Mode](/P/norm-mode)** <br>
-   &emsp;&ensp; 3.2.11. **[Variance](/P/norm-var)** <br>
-   &emsp;&ensp; 3.2.12. **[Full width at half maximum](/P/norm-fwhm)** <br>
-   &emsp;&ensp; 3.2.13. **[Differential entropy](/P/norm-dent)** <br>
-   &emsp;&ensp; 3.2.14. **[Moment-generating function](/P/norm-mgf)** <br>
+   &emsp;&ensp; 3.2.5. **[Moment-generating function](/P/norm-mgf)** <br>
+   &emsp;&ensp; 3.2.6. **[Cumulative distribution function](/P/norm-cdf)** <br>
+   &emsp;&ensp; 3.2.7. **[Cumulative distribution function without error function](/P/norm-cdfwerf)** <br>
+   &emsp;&ensp; 3.2.8. **[Quantile function](/P/norm-qf)** <br>
+   &emsp;&ensp; 3.2.9. **[Mean](/P/norm-mean)** <br>
+   &emsp;&ensp; 3.2.10. **[Median](/P/norm-med)** <br>
+   &emsp;&ensp; 3.2.11. **[Mode](/P/norm-mode)** <br>
+   &emsp;&ensp; 3.2.12. **[Variance](/P/norm-var)** <br>
+   &emsp;&ensp; 3.2.13. **[Full width at half maximum](/P/norm-fwhm)** <br>
+   &emsp;&ensp; 3.2.14. **[Differential entropy](/P/norm-dent)** <br>
 
    3.3. Gamma distribution <br>
    &emsp;&ensp; 3.3.1. *[Definition](/D/gam)* <br>
@@ -264,6 +264,9 @@ title: "Table of Contents"
    3.6. Wald distribution <br>
    &emsp;&ensp; 3.6.1. *[Definition](/D/wald)* <br>
    &emsp;&ensp; 3.6.2. **[Probability density function](/P/wald-pdf)** <br>
+   &emsp;&ensp; 3.6.3. **[Moment-generating function](/P/wald-mgf)** <br>
+   &emsp;&ensp; 3.6.4. **[Mean](/P/wald-mean)** <br>
+   &emsp;&ensp; 3.6.5. **[Variance](/P/wald-var)** <br>   
 
 4. Multivariate continuous distributions
 

P/momcent-1st.md

@@ -5,7 +5,7 @@ mathjax: true
 author: "Joram Soch"
 affiliation: "BCCN Berlin"
 e_mail: "joram.soch@bccn-berlin.de"
-date: 2020-08-19 07:51:00
+date: 2020-09-09 07:51:00
 
 title: "First central moment is zero"
 chapter: "General Theorems"

P/wald-mean.md

@@ -20,43 +20,48 @@ shortcut: "wald-mean"
 username: "tomfaulkenberry"
 ---
 
+
 **Theorem:** Let $X$ be a positive [random variable](/D/rvar) following a [Wald distribution](/D/wald):
 
 $$ \label{eq:wald}
 X \sim \mathrm{Wald}(\gamma, \alpha) \; .
 $$
 
-
 Then, the [mean or expected value](/D/mean) of $X$ is
 
-
 $$ \label{eq:wald-mean}
 \mathrm{E}(X) = \frac{\alpha}{\gamma} \; .
 $$
 
 
-**Proof:** The mean or expected value $\mathrm{E}(X)$ is the first [moment](/D/mom) of $X$, so we can use the [moment-generating function of the Wald distribution](/P/wald-mgf) to calculate
+**Proof:** The mean or expected value $\mathrm{E}(X)$ is the first [moment](/D/mom) of $X$, so [we can use](/P/mom-mgf) the [moment-generating function of the Wald distribution](/P/wald-mgf) to calculate
 
 $$ \label{eq:wald-moment}
 \mathrm{E}(X) = M_X'(0) \; .
 $$
 
-First we differentiate $M_X(t) = \exp\left[\alpha \gamma - \sqrt{\alpha^2(\gamma^2-2t)}\right]$ with respect to $t$. Using the chain rule gives
+First we differentiate
+
+$$ \label{eq:wald-mgf}
+M_X(t) = \exp\left[\alpha \gamma - \sqrt{\alpha^2(\gamma^2-2t)}\right]
+$$
+
+with respect to $t$. Using the chain rule gives
 
 $$ \label{eq:wald-mean-s1}
 \begin{split}
   M_X'(t) &= \exp\left[\alpha \gamma - \sqrt{\alpha^2(\gamma^2-2t)}\right] \cdot -\frac{1}{2}\left(\alpha^2(\gamma^2-2t)\right)^{-1/2}\cdot -2\alpha^2 \\
-  &= \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \cdot \frac{\alpha^2}{\sqrt{\alpha^2(\gamma^2-2t)}} \\
+  &= \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \cdot \frac{\alpha^2}{\sqrt{\alpha^2(\gamma^2-2t)}} \; .
 \end{split}
 $$
 
-Evaluating \eqref{wald-mean-s1} at $t=0$ gives the desired result:
+Evaluating \eqref{eq:wald-mean-s1} at $t=0$ gives the desired result:
 
 $$ \label{eq:wald-mean-s2}
 \begin{split}
-  M_X'(0) &= \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2(0))}\right] \cdot \frac{\alpha^2}{\sqrt{\alpha^2(\gamma^2-2(0))}}\\
-          &= \exp\left[\alpha \gamma - \sqrt{\alpha^2 \cdot \gamma^2}\right]\cdot \frac{\alpha^2}{\sqrt{\alpha^2\cdot \gamma^2}}\\
-          &= \exp[0] \cdot \frac{\alpha^2}{\alpha \gamma}\\
-  &= \frac{\alpha}{\gamma}\\ \; .
+  M_X'(0) &= \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2(0))}\right] \cdot \frac{\alpha^2}{\sqrt{\alpha^2(\gamma^2-2(0))}} \\
+          &= \exp\left[\alpha \gamma - \sqrt{\alpha^2 \cdot \gamma^2}\right]\cdot \frac{\alpha^2}{\sqrt{\alpha^2\cdot \gamma^2}} \\
+          &= \exp[0] \cdot \frac{\alpha^2}{\alpha \gamma} \\
+          &= \frac{\alpha}{\gamma} \; .
 \end{split}
 $$

P/wald-mgf.md

@@ -17,10 +17,9 @@ sources:
   - authors: "Siegrist, K."
     year: 2020
     title: "The Wald Distribution"
-    in: "Random (formerly Virtual Laboratories in Probability and Statistics)"
+    in: "Random: Probability, Mathematical Statistics, Stochastic Processes"
     pages: "retrieved on 2020-09-13"
     url: "https://www.randomservices.org/random/special/Wald.html"
-
   - authors: "National Institute of Standards and Technology"
     year: 2020
     title: "NIST Digital Library of Mathematical Functions"
@@ -39,12 +38,10 @@ $$ \label{eq:wald}
 X \sim \mathrm{Wald}(\gamma, \alpha) \; .
 $$
 
-
 Then, the [moment-generating function](/D/mgf) of $X$ is
 
-
 $$ \label{eq:wald-mgf}
-M_X(t) = \exp \left[ \alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right].
+M_X(t) = \exp \left[ \alpha\gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \; .
 $$
 
 
@@ -64,8 +61,8 @@ Using the definition of [expected value for continuous random variables](/D/mean
 
 $$ \label{eq:wald-mgf-s1}
 \begin{split}
-M_X(t) &= \int_0^{\infty} e^{tx} \cdot \frac{\alpha}{\sqrt{2\pi x^3}}\cdot \exp\left[-\frac{(\alpha-\gamma x)^2}{2x}\right]dx\\
-&= \frac{\alpha}{\sqrt{2\pi}}\int_0^{\infty} x^{-3/2}\cdot \exp\left[tx - \frac{(\alpha-\gamma x)^2}{2x}\right]dx\\
+M_X(t) &= \int_0^{\infty} e^{tx} \cdot \frac{\alpha}{\sqrt{2\pi x^3}}\cdot \exp\left[-\frac{(\alpha-\gamma x)^2}{2x}\right]dx \\
+&= \frac{\alpha}{\sqrt{2\pi}}\int_0^{\infty} x^{-3/2}\cdot \exp\left[tx - \frac{(\alpha-\gamma x)^2}{2x}\right]dx \; .
 \end{split}
 $$
 
@@ -91,9 +88,9 @@ Starting from \eqref{eq:wald-mgf-s1}, we can expand the binomial term and rearra
 
 $$ \label{eq:wald-mgf-s2}
 \begin{split}
-M_X(t) &= \frac{\alpha}{\sqrt{2\pi}} \int_0^{\infty} x^{-3/2}\cdot \exp\left[ tx - \frac{\alpha^2}{2x} + \alpha\gamma - \frac{\gamma^2x}{2}\right]dx\\
-       &= \frac{\alpha}{\sqrt{2\pi}}\cdot e^{\alpha \gamma} \int_0^{\infty} x^{-3/2}\cdot \exp\left[\left(t-\frac{\gamma^2}{2}\right)x - \frac{\alpha^2}{2x}\right]dx\\
-  &= \frac{\alpha}{\sqrt{2\pi}}\cdot e^{\alpha \gamma} \int_0^{\infty} x^{-3/2}\cdot \exp \left[-\frac{1}{2}\left(\gamma^2-2t\right)x - \frac{1}{2}\cdot \frac{\alpha^2}{x}\right]dx\\ \; .
+M_X(t) &= \frac{\alpha}{\sqrt{2\pi}} \int_0^{\infty} x^{-3/2}\cdot \exp\left[ tx - \frac{\alpha^2}{2x} + \alpha\gamma - \frac{\gamma^2x}{2}\right]dx \\
+       &= \frac{\alpha}{\sqrt{2\pi}}\cdot e^{\alpha \gamma} \int_0^{\infty} x^{-3/2}\cdot \exp\left[\left(t-\frac{\gamma^2}{2}\right)x - \frac{\alpha^2}{2x}\right]dx \\
+       &= \frac{\alpha}{\sqrt{2\pi}}\cdot e^{\alpha \gamma} \int_0^{\infty} x^{-3/2}\cdot \exp \left[-\frac{1}{2}\left(\gamma^2-2t\right)x - \frac{1}{2}\cdot \frac{\alpha^2}{x}\right]dx \; .
 \end{split}
 $$
 
@@ -107,10 +104,10 @@ Combining with \eqref{eq:bessel-fact1} and simplifying gives
 
 $$ \label{eq:wald-mgf-s4}
 \begin{split}
-  M_X(t) &= \frac{\alpha}{\sqrt{2\pi}}\cdot e^{\alpha \gamma}\cdot 2\left(\frac{\gamma^2-2t}{\alpha^2}\right)^{1/4} \cdot \sqrt{\frac{\pi}{2\sqrt{\alpha^2(\gamma^2-2t)}}}\cdot \exp\left[-\sqrt{\alpha^2(\gamma^2-2t)}\right]\\
-         &= \frac{\alpha}{\sqrt{2}\cdot \sqrt{\pi}}\cdot e^{\alpha \gamma}\cdot 2 \cdot \frac{(\gamma^2-2t)^{1/4}}{\sqrt{\alpha}}\cdot \frac{\sqrt{\pi}}{\sqrt{2}\cdot \sqrt{\alpha}\cdot (\gamma^2-2t)^{1/4}}\cdot \exp\left[-\sqrt{\alpha^2(\gamma^2-2t)}\right]\\
-         &= e^{\alpha \gamma} \cdot \exp\left[-\sqrt{\alpha^2(\gamma^2-2t)}\right]\\
-  &= \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\\ \; .
+  M_X(t) &= \frac{\alpha}{\sqrt{2\pi}}\cdot e^{\alpha \gamma}\cdot 2\left(\frac{\gamma^2-2t}{\alpha^2}\right)^{1/4} \cdot \sqrt{\frac{\pi}{2\sqrt{\alpha^2(\gamma^2-2t)}}}\cdot \exp\left[-\sqrt{\alpha^2(\gamma^2-2t)}\right] \\
+         &= \frac{\alpha}{\sqrt{2}\cdot \sqrt{\pi}}\cdot e^{\alpha \gamma}\cdot 2 \cdot \frac{(\gamma^2-2t)^{1/4}}{\sqrt{\alpha}}\cdot \frac{\sqrt{\pi}}{\sqrt{2}\cdot \sqrt{\alpha}\cdot (\gamma^2-2t)^{1/4}}\cdot \exp\left[-\sqrt{\alpha^2(\gamma^2-2t)}\right] \\
+         &= e^{\alpha \gamma} \cdot \exp\left[-\sqrt{\alpha^2(\gamma^2-2t)}\right] \\
+         &= \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \; .
 \end{split}
 $$
 

P/wald-var.md

@@ -20,16 +20,15 @@ shortcut: "wald-var"
 username: "tomfaulkenberry"
 ---
 
+
 **Theorem:** Let $X$ be a positive [random variable](/D/rvar) following a [Wald distribution](/D/wald):
 
 $$ \label{eq:wald}
 X \sim \mathrm{Wald}(\gamma, \alpha) \; .
 $$
 
-
 Then, the [variance](/D/var) of $X$ is
 
-
 $$ \label{eq:wald-var}
 \mathrm{Var}(X) = \frac{\alpha}{\gamma^3} \; .
 $$
@@ -47,13 +46,19 @@ $$ \label{eq:wald-moment}
 \mathrm{E}(X^2) = M_X''(0) \; .
 $$
 
-First we differentiate $M_X(t) = \exp\left[\alpha \gamma - \sqrt{\alpha^2(\gamma^2-2t)}\right]$ with respect to $t$. Using the chain rule gives
+First we differentiate
+
+$$ \label{eq:wald-mgf}
+M_X(t) = \exp\left[\alpha \gamma - \sqrt{\alpha^2(\gamma^2-2t)}\right]
+$$
+
+with respect to $t$. Using the chain rule gives
 
 $$ \label{eq:wald-var-s1}
 \begin{split}
   M_X'(t) &= \exp\left[\alpha \gamma - \sqrt{\alpha^2(\gamma^2-2t)}\right] \cdot -\frac{1}{2}\left(\alpha^2(\gamma^2-2t)\right)^{-1/2}\cdot -2\alpha^2 \\
           &= \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right] \cdot \frac{\alpha^2}{\sqrt{\alpha^2(\gamma^2-2t)}} \\
-  &= \alpha \cdot \exp\left[\alpha \gamma -\sqrt{\alpha^2(\gamma^2-2t)}\right] \cdot (\gamma^2-2t)^{-1/2}\\ \; .
+          &= \alpha \cdot \exp\left[\alpha \gamma -\sqrt{\alpha^2(\gamma^2-2t)}\right] \cdot (\gamma^2-2t)^{-1/2} \; .
 \end{split}
 $$
 
@@ -62,31 +67,31 @@ Now we use the product rule to obtain the second derivative:
 $$ \label{eq:wald-var-s2}
 \begin{split}
   M_X''(t) &= \alpha \cdot \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\cdot (\gamma^2-2t)^{-1/2}\cdot -\frac{1}{2}\left(\alpha^2(\gamma^2-2t)\right)^{-1/2}\cdot -2\alpha^2 \\
-           &+ \alpha \cdot \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\cdot -\frac{1}{2}(\gamma^2-2t)^{-3/2}\cdot -2\\
-           &= \alpha^2\cdot \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\cdot (\gamma^2-2t)^{-1}\\
-           &+ \alpha\cdot \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\cdot (\gamma^2-2t)^{-3/2}\\
-  &= \alpha \cdot \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\left[\frac{\alpha}{\gamma^2-2t}+\frac{1}{\sqrt{(\gamma^2-2t)^3}}\right]\\ \; .
+           &+ \alpha \cdot \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\cdot -\frac{1}{2}(\gamma^2-2t)^{-3/2}\cdot -2 \\
+           &= \alpha^2\cdot \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\cdot (\gamma^2-2t)^{-1} \\
+           &+ \alpha\cdot \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\cdot (\gamma^2-2t)^{-3/2} \\
+           &= \alpha \cdot \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2t)}\right]\left[\frac{\alpha}{\gamma^2-2t}+\frac{1}{\sqrt{(\gamma^2-2t)^3}}\right] \; .
 \end{split}
 $$
 
 Applying \eqref{eq:wald-moment} yields
 
 $$ \label{eq:wald-var-s3}
 \begin{split}
-  \mathrm{E}(X^2) &= M_X''(0)\\
-                  &= \alpha \cdot \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2(0))}\right]\left[\frac{\alpha}{\gamma^2-2(0)}+\frac{1}{\sqrt{(\gamma^2-2(0))^3}}\right]\\
-                  &= \alpha \cdot \exp\left[\alpha \gamma - \alpha \gamma\right] \cdot \left[\frac{\alpha}{\gamma^2} + \frac{1}{\gamma^3}\right]\\
-  &= \frac{\alpha^2}{\gamma^2} + \frac{\alpha}{\gamma^3}\\ \; .
+  \mathrm{E}(X^2) &= M_X''(0) \\
+                  &= \alpha \cdot \exp\left[\alpha \gamma-\sqrt{\alpha^2(\gamma^2-2(0))}\right]\left[\frac{\alpha}{\gamma^2-2(0)}+\frac{1}{\sqrt{(\gamma^2-2(0))^3}}\right] \\
+                  &= \alpha \cdot \exp\left[\alpha \gamma - \alpha \gamma\right] \cdot \left[\frac{\alpha}{\gamma^2} + \frac{1}{\gamma^3}\right] \\
+                  &= \frac{\alpha^2}{\gamma^2} + \frac{\alpha}{\gamma^3} \; .
 \end{split}
 $$
 
 Since the [mean of a Wald distribution](/P/wald-mean) is given by $\mathrm{E}(X)=\alpha/\gamma$, we can apply \eqref{eq:var-mean} to show
 
 $$ \label{eq:wald-var-s4}
 \begin{split}
-  \mathrm{Var}(X) &= \mathrm{E}(X^2) - \mathrm{E}(X)^2\\
-                  &= \frac{\alpha^2}{\gamma^2} + \frac{\alpha}{\gamma^3} - \left(\frac{\alpha}{\gamma}\right)^2\\
-                  &= \frac{\alpha}{\gamma^3}\\
+  \mathrm{Var}(X) &= \mathrm{E}(X^2) - \mathrm{E}(X)^2 \\
+                  &= \frac{\alpha^2}{\gamma^2} + \frac{\alpha}{\gamma^3} - \left(\frac{\alpha}{\gamma}\right)^2 \\
+                  &= \frac{\alpha}{\gamma^3}
 \end{split}
 $$