add wald and wald-pdf

Author: tomfaulkenberry

D/wald.md

@@ -0,0 +1,43 @@
+---
+layout: definition
+mathjax: true
+
+author: "Thomas J. Faulkenberry"
+affiliation: "Tarleton State University"
+e_mail: "faulkenberry@tarleton.edu"
+date: 2020-09-04 12:00:00
+
+title: "Wald distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Wald distribution"
+definition: "Definition"
+
+sources:
+  - authors: "Anders, R., Alario, F. -X., and van Maanen, L."
+    year: 2016
+    title: "The Shifted Wald Distribution for Response Time Data Analysis"
+    in: "Psychological Methods"
+    pages: "vol. 21, no. 3, pp. 309-327"
+    url: "https://dx.doi.org/10.1037/met0000066"
+    doi: "10.1037/met0000066"
+
+def_id: "D95"
+shortcut: "wald"
+username: "tomfaulkenberry"
+---
+
+
+**Definition**: Let $X$ be a [random variable](/D/rvar). Then, $X$ is said to follow a Wald distribution with drift rate $\gamma$ and threshold $\alpha$
+
+$$ \label{eq:wald}
+X \sim \mathrm{Wald}(\gamma, \alpha) \; ,
+$$
+
+if and only if its [probability density function](/D/pdf) is given by
+
+$$ \label{eq:wald-pdf}
+\mathrm{Wald}(x; \gamma, \alpha) = \frac{\alpha}{\sqrt{2\pi x^3}}\exp\Bigl(-\frac{(\alpha-\gamma x)^2}{2x}\Bigr)
+$$
+
+where $\gamma > 0$, $\alpha > 0$, and the density is zero if $x \leq 0$.

P/wald-pdf.md

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+---
+layout: proof
+mathjax: true
+
+author: "Thomas J. Faulkenberry"
+affiliation: "Tarleton State University"
+e_mail: "faulkenberry@tarleton.edu"
+date: 2020-09-04 12:00:00
+
+title: "Probability density function of the Wald distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Wald distribution"
+theorem: "Probability density function"
+
+sources:
+
+proof_id: "P162"
+shortcut: "wald"
+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 [probability density function](/D/pdf) of $X$ is
+
+$$ \label{eq:wald-pdf}
+f_X(x) = \frac{\alpha}{\sqrt{2\pi x^3}}\exp\Bigl(-\frac{(\alpha-\gamma x)^2}{2x}\Bigr) \; .
+$$
+
+
+**Proof:** This follows directly from the [definition of the Wald distribution](/D/wald).