Merge pull request #38 from StatProofBook/master

JoramSoch · web-flow · commit 6f79e43201f0 · 2020-11-06T05:34:59.000+01:00
update to master
diff --git a/D/ind.md b/D/ind.md
@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Generally speaking, [random variables](/D/rvar) are statistically independent, if their [joint probability](/D/prob-joint) can be expressed in terms of their [marginal probability](/D/prob-marg).
+**Definition:** Generally speaking, [random variables](/D/rvar) are statistically independent, if their [joint probability](/D/prob-joint) can be expressed in terms of their [marginal probabilities](/D/prob-marg).
 
 <br>
 1) A set of discrete [random variables](/D/rvar) $X_1, \ldots, X_n$ with possible values $\mathcal{X}_1, \ldots, \mathcal{X}_n$ is called statistically independent, if
@@ -45,7 +45,7 @@ $$ \label{eq:cont-ind-F}
 F_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = \prod_{i=1}^{n} F_{X_i}(x_i) \quad \text{for all} \; x_i \in \mathcal{X}_i, \; i = 1, \ldots, n
 $$
 
-or equivalently, if the [probability densities](/D/pdf) exist,
+or equivalently, if the [probability densities](/D/pdf) exist, if
 
 $$ \label{eq:cont-ind-f}
 f_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = \prod_{i=1}^{n} f_{X_i}(x_i) \quad \text{for all} \; x_i \in \mathcal{X}_i, \; i = 1, \ldots, n
diff --git a/D/pdf.md b/D/pdf.md
@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let $X$ be a continuous [random variable](/D/rvar) with possible outcomes $\mathcal{X}$. Then, $f_X(x): \mathbb{R} \to \mathbb{R}$ is the probability density function of $X$, if
+**Definition:** Let $X$ be a [continuous](/D/rvar-disc) [random variable](/D/rvar) with possible outcomes $\mathcal{X}$. Then, $f_X(x): \mathbb{R} \to \mathbb{R}$ is the probability density function of $X$, if
 
 $$ \label{eq:pdf-def-s0}
 f_X(x) \geq 0
diff --git a/D/pmf.md b/D/pmf.md
@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let $X$ be a discrete [random variable](/D/rvar) with possible outcomes $\mathcal{X}$. Then, $f_X(x): \mathbb{R} \to [0,1]$ is the probability mass function of $X$, if
+**Definition:** Let $X$ be a [discrete](/D/rvar-disc) [random variable](/D/rvar) with possible outcomes $\mathcal{X}$. Then, $f_X(x): \mathbb{R} \to [0,1]$ is the probability mass function of $X$, if
 
 $$ \label{eq:pmf-def-s0}
 f_X(x) = 0
diff --git a/I/Definition_by_Author.md b/I/Definition_by_Author.md
@@ -4,7 +4,7 @@ title: "Definition by Author"
 ---
 
 
-### JoramSoch (100 definitions)
+### JoramSoch (101 definitions)
 
 - [Akaike information criterion](/D/aic)
 - [Bayesian information criterion](/D/bic)
@@ -34,6 +34,7 @@ title: "Definition by Author"
 - [Differential entropy](/D/dent)
 - [Dirichlet distribution](/D/dir)
 - [Dirichlet-distributed data](/D/dir-data)
+- [Discrete and continuous random variable](/D/rvar-disc)
 - [Discrete uniform distribution](/D/duni)
 - [Estimation matrix](/D/emat)
 - [Exceedance probability](/D/prob-exc)
diff --git a/I/Definition_by_Number.md b/I/Definition_by_Number.md
@@ -111,3 +111,4 @@ title: "Definition by Number"
 | D102 | mode | [Mode](/D/mode) | JoramSoch | 2020-10-15 |
 | D103 | prob-exc | [Exceedance probability](/D/prob-exc) | JoramSoch | 2020-10-22 |
 | D104 | dir-data | [Dirichlet-distributed data](/D/dir-data) | JoramSoch | 2020-10-22 |
+| D105 | rvar-disc | [Discrete and continuous random variable](/D/rvar-disc) | JoramSoch | 2020-10-29 |
diff --git a/I/Definition_by_Topic.md b/I/Definition_by_Topic.md
@@ -46,6 +46,7 @@ title: "Definition by Topic"
 - [Differential entropy](/D/dent)
 - [Dirichlet distribution](/D/dir)
 - [Dirichlet-distributed data](/D/dir-data)
+- [Discrete and continuous random variable](/D/rvar-disc)
 - [Discrete uniform distribution](/D/duni)
 
 ### E
diff --git a/I/Proof_by_Author.md b/I/Proof_by_Author.md
@@ -4,7 +4,7 @@ title: "Proof by Author"
 ---
 
 
-### JoramSoch (173 proofs)
+### JoramSoch (176 proofs)
 
 - [(Non-)Multiplicativity of the expected value](/P/mean-mult)
 - [Additivity of the Kullback-Leibler divergence for independent distributions](/P/kl-add)
@@ -25,6 +25,7 @@ title: "Proof by Author"
 - [Convexity of the Kullback-Leibler divergence](/P/kl-conv)
 - [Convexity of the cross-entropy](/P/entcross-conv)
 - [Covariance of independent random variables](/P/cov-ind)
+- [Cumulative distribution function of a strictly increasing function of a random variable](/P/cdf-sifct)
 - [Cumulative distribution function of the continuous uniform distribution](/P/cuni-cdf)
 - [Cumulative distribution function of the discrete uniform distribution](/P/duni-cdf)
 - [Cumulative distribution function of the exponential distribution](/P/exp-cdf)
@@ -127,6 +128,7 @@ title: "Proof by Author"
 - [Posterior model probability in terms of log Bayes factor](/P/pmp-lbf)
 - [Posterior probability of the alternative hypothesis for Bayesian linear regression](/P/blr-pp)
 - [Probability and log-odds in logistic regression](/P/logreg-pnlo)
+- [Probability density function of a strictly increasing function of a continuous random variable](/P/pdf-sifct)
 - [Probability density function of the Dirichlet distribution](/P/dir-pdf)
 - [Probability density function of the beta distribution](/P/beta-pdf)
 - [Probability density function of the continuous uniform distribution](/P/cuni-pdf)
@@ -136,6 +138,7 @@ title: "Proof by Author"
 - [Probability density function of the multivariate normal distribution](/P/mvn-pdf)
 - [Probability density function of the normal distribution](/P/norm-pdf)
 - [Probability density function of the normal-gamma distribution](/P/ng-pdf)
+- [Probability mass function of a strictly increasing function of a discrete random variable](/P/pmf-sifct)
 - [Probability mass function of the Bernoulli distribution](/P/bern-pmf)
 - [Probability mass function of the Poisson distribution](/P/poiss-pmf)
 - [Probability mass function of the binomial distribution](/P/bin-pmf)
diff --git a/I/Proof_by_Number.md b/I/Proof_by_Number.md
@@ -189,3 +189,6 @@ title: "Proof by Number"
 | P180 | norm-snorm3 | [Relation between normal distribution and standard normal distribution](/P/norm-snorm3) | JoramSoch | 2020-10-22 |
 | P181 | dir-ep | [Exceedance probabilities for the the Dirichlet distribution](/P/dir-ep) | JoramSoch | 2020-10-22 |
 | P182 | dir-mle | [Maximum likelihood estimation for Dirichlet-distributed data](/P/dir-mle) | JoramSoch | 2020-10-22 |
+| P183 | cdf-sifct | [Cumulative distribution function of a strictly increasing function of a random variable](/P/cdf-sifct) | JoramSoch | 2020-10-29 |
+| P184 | pmf-sifct | [Probability mass function of a strictly increasing function of a discrete random variable](/P/pmf-sifct) | JoramSoch | 2020-10-29 |
+| P185 | pdf-sifct | [Probability density function of a strictly increasing function of a continuous random variable](/P/pdf-sifct) | JoramSoch | 2020-10-29 |
diff --git a/I/Proof_by_Topic.md b/I/Proof_by_Topic.md
@@ -35,6 +35,7 @@ title: "Proof by Topic"
 - [Convexity of the Kullback-Leibler divergence](/P/kl-conv)
 - [Convexity of the cross-entropy](/P/entcross-conv)
 - [Covariance of independent random variables](/P/cov-ind)
+- [Cumulative distribution function of a strictly increasing function of a random variable](/P/cdf-sifct)
 - [Cumulative distribution function of the continuous uniform distribution](/P/cuni-cdf)
 - [Cumulative distribution function of the discrete uniform distribution](/P/duni-cdf)
 - [Cumulative distribution function of the exponential distribution](/P/exp-cdf)
@@ -177,6 +178,7 @@ title: "Proof by Topic"
 - [Posterior model probability in terms of log Bayes factor](/P/pmp-lbf)
 - [Posterior probability of the alternative hypothesis for Bayesian linear regression](/P/blr-pp)
 - [Probability and log-odds in logistic regression](/P/logreg-pnlo)
+- [Probability density function of a strictly increasing function of a continuous random variable](/P/pdf-sifct)
 - [Probability density function of the Dirichlet distribution](/P/dir-pdf)
 - [Probability density function of the Wald distribution](/P/wald-pdf)
 - [Probability density function of the beta distribution](/P/beta-pdf)
@@ -187,6 +189,7 @@ title: "Proof by Topic"
 - [Probability density function of the multivariate normal distribution](/P/mvn-pdf)
 - [Probability density function of the normal distribution](/P/norm-pdf)
 - [Probability density function of the normal-gamma distribution](/P/ng-pdf)
+- [Probability mass function of a strictly increasing function of a discrete random variable](/P/pmf-sifct)
 - [Probability mass function of the Bernoulli distribution](/P/bern-pmf)
 - [Probability mass function of the Poisson distribution](/P/poiss-pmf)
 - [Probability mass function of the binomial distribution](/P/bin-pmf)
diff --git a/I/Proofs_without_Source.md b/I/Proofs_without_Source.md
@@ -9,7 +9,6 @@ title: "Proofs without Source"
 - [Cumulative distribution function of the continuous uniform distribution](/P/cuni-cdf)
 - [Cumulative distribution function of the discrete uniform distribution](/P/duni-cdf)
 - [Cumulative distribution function of the exponential distribution](/P/exp-cdf)
-- [Cumulative distribution function of the gamma distribution](/P/gam-cdf)
 - [Derivation of Bayesian model averaging](/P/bma-der)
 - [Derivation of the log Bayes factor](/P/lbf-der)
 - [Derivation of the log family evidence](/P/lfe-der)
diff --git a/P/gam-cdf.md b/P/gam-cdf.md
@@ -14,6 +14,12 @@ topic: "Gamma distribution"
 theorem: "Cumulative distribution function"
 
 sources:
+  - authors: "Wikipedia"
+    year: 2020
+    title: "Incomplete gamma function"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2020-10-29"
+    url: "https://en.wikipedia.org/wiki/Incomplete_gamma_function#Definition"
 
 proof_id: "P178"
 shortcut: "gam-cdf"
diff --git a/P/gam-sgam.md b/P/gam-sgam.md
@@ -66,21 +66,21 @@ $$
 
 Applying \eqref{eq:cdf-sifct} to \eqref{eq:gam-cdf}, we have:
 
-\begin{equation} \label{eq:Y-cdf-s1}
+$$ \label{eq:Y-cdf-s1}
 \begin{split}
 F_Y(y) &\overset{\eqref{eq:cdf-sifct}}{=} F_X(g^{-1}(y)) \\
 &\overset{\eqref{eq:gam-cdf}}{=} \int_{-\infty}^{y/b} \frac{b^a}{\Gamma(a)} t^{a-1} \exp[-b t] \, \mathrm{d}t \; .
 \end{split}
-\end{equation}
+$$
 
 Substituting $s = b t$, such that $t = s/b$, we obtain
 
-\begin{equation} \label{eq:Z-cdf-s2}
+$$ \label{eq:Z-cdf-s2}
 \begin{split}
 F_Y(y) &= \int_{-b \infty}^{b (y/b)} \frac{b^a}{\Gamma(a)} \left(\frac{s}{b}\right)^{a-1} \exp\left[-b \left(\frac{s}{b}\right)\right] \, \mathrm{d}\left(\frac{s}{b}\right) \\
 &= \int_{-\infty}^{y} \frac{b^a}{\Gamma(a)} \, \frac{1}{b^{a-1} \, b} \, s^{a-1} \exp[-s] \, \mathrm{d}s \\
 &= \int_{-\infty}^{y} \frac{1}{\Gamma(a)} s^{a-1} \exp[-s] \, \mathrm{d}s
 \end{split}
-\end{equation}
+$$
 
 which is the [cumulative distribution function](/D/cdf) of the [standard gamma distribution](/D/sgam).
diff --git a/P/pdf-sifct.md b/P/pdf-sifct.md
@@ -73,12 +73,12 @@ $$
 
 2) If $y$ belongs to the support of $Y$, then $f_Y(y)$ can be derived using the chain rule:
 
-\begin{equation} \label{eq:pdf-sifct-p2}
+$$ \label{eq:pdf-sifct-p2}
 \begin{split}
 f_Y(y) &\overset{\eqref{eq:pdf-cdf}}{=} \frac{\mathrm{d}}{\mathrm{d}y} F_Y(y) \\
 &\overset{\eqref{eq:cdf-sifct}}{=} \frac{\mathrm{d}}{\mathrm{d}y} F_X(g^{-1}(y)) \\
 &= f_X(g^{-1}(y)) \, \frac{\mathrm{d}g^{-1}(y)}{\mathrm{d}y} \; .
 \end{split}
-\end{equation}
+$$
 
 Taking together \eqref{eq:pdf-sifct-p1} and \eqref{eq:pdf-sifct-p2}, eventually proves \eqref{eq:pdf-sifct}.
