Merge pull request #41 from StatProofBook/master

JoramSoch · web-flow · commit 3fb0f451fa00 · 2020-11-25T08:18:14.000+01:00
update to master
diff --git a/D/ind-cond.md b/D/ind-cond.md
@@ -39,7 +39,7 @@ $$
 where $p(x_1, \ldots, x_n \vert y)$ are the [joint (conditional) probabilities](/D/prob-joint) of $X_1, \ldots, X_n$ given $Y$ and $p(x_i)$ are the [marginal (conditional) probabilities](/D/prob-marg) of $X_i$ given $Y$.
 
 <br>
-2) A set of [random variables](/D/rvar) $X_1, \ldots, X_n$ with possible values $\mathcal{X}_1, \ldots, \mathcal{X}_n$ is called conditionally independent given the random variable $Y$ with possible values $\mathcal{Y}$, if
+2) A set of [continuous random variables](/D/rvar-disc) $X_1, \ldots, X_n$ with possible values $\mathcal{X}_1, \ldots, \mathcal{X}_n$ is called conditionally independent given the random variable $Y$ with possible values $\mathcal{Y}$, if
 
 $$ \label{eq:cond-ind-F}
 F_{X_1,\ldots,X_n|Y=y}(x_1,\ldots,x_n) = \prod_{i=1}^{n} F_{X_i|Y=y}(x_i) \quad \text{for all} \; x_i \in \mathcal{X}_i \quad \text{and all} \; y \in \mathcal{Y}
diff --git a/D/qf.md b/D/qf.md
@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition:** Let $X$ be a [random variable](/D/rvar) with the [cumulative distribution function](/D/cdf) (CDF) $F_X(x)$. Then, the function $Q_X(p): [0,1] \to \mathbb{R}$ which is the inverse CDF is the quantile function (QF) of $X$. More precisly, the QF is the function that, for a given quantile $p \in [0,1]$, returns the smallest $x$ for which $F_X(x) = p$:
+**Definition:** Let $X$ be a [random variable](/D/rvar) with the [cumulative distribution function](/D/cdf) (CDF) $F_X(x)$. Then, the function $Q_X(p): [0,1] \to \mathbb{R}$ which is the inverse CDF is the quantile function (QF) of $X$. More precisely, the QF is the function that, for a given quantile $p \in [0,1]$, returns the smallest $x$ for which $F_X(x) = p$:
 
 $$ \label{eq:qf}
 Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .
diff --git a/D/rvar.md b/D/rvar.md
@@ -33,4 +33,4 @@ username: "JoramSoch"
 
 * formally, as a [measurable function](/D/meas-fct) $X$ defined on a [probability space](/D/prob-spc) $(\Omega, \mathcal{F}, P)$ that maps from a sample space $\Omega$ to the real numbers $\mathbb{R}$ using an event space $\mathcal{F}$ and a [probability function](/D/pmf) $P$;
 
-* more broadly, as any random quantity $X$ such as a [random scalar](/D/rvar), a [random vector](/D/rvec) or a [random matrix](/D/rmat).
+* more broadly, as any random quantity $X$ such as a [random event](/D/reve), a [random scalar](/D/rvar), a [random vector](/D/rvec) or a [random matrix](/D/rmat).
diff --git a/I/Definition_by_Author.md b/I/Definition_by_Author.md
@@ -4,7 +4,7 @@ title: "Definition by Author"
 ---
 
 
-### JoramSoch (104 definitions)
+### JoramSoch (108 definitions)
 
 - [Akaike information criterion](/D/aic)
 - [Bayesian information criterion](/D/bic)
@@ -19,6 +19,7 @@ title: "Definition by Author"
 - [Coefficient of determination](/D/rsq)
 - [Conditional differential entropy](/D/dent-cond)
 - [Conditional entropy](/D/ent-cond)
+- [Conditional independence](/D/ind-cond)
 - [Conditional probability distribution](/D/dist-cond)
 - [Constant](/D/const)
 - [Continuous uniform distribution](/D/cuni)
@@ -27,6 +28,7 @@ title: "Definition by Author"
 - [Covariance](/D/cov)
 - [Covariance matrix](/D/covmat)
 - [Cross-entropy](/D/ent-cross)
+- [Cross-validated log model evidence](/D/cvlme)
 - [Cumulant-generating function](/D/cgf)
 - [Cumulative distribution function](/D/cdf)
 - [Deviance information criterion](/D/dic)
@@ -93,6 +95,8 @@ title: "Definition by Author"
 - [Probability-generating function](/D/pgf)
 - [Projection matrix](/D/pmat)
 - [Quantile function](/D/qf)
+- [Random event](/D/reve)
+- [Random experiment](/D/rexp)
 - [Random matrix](/D/rmat)
 - [Random variable](/D/rvar)
 - [Random vector](/D/rvec)
diff --git a/I/Definition_by_Number.md b/I/Definition_by_Number.md
@@ -115,3 +115,7 @@ title: "Definition by Number"
 | D106 | rvar-uni | [Univariate and multivariate random variable](/D/rvar-uni) | JoramSoch | 2020-11-06 |
 | D107 | min | [Minimum](/D/min) | JoramSoch | 2020-11-12 |
 | D108 | max | [Maximum](/D/max) | JoramSoch | 2020-11-12 |
+| D109 | rexp | [Random experiment](/D/rexp) | JoramSoch | 2020-11-19 |
+| D110 | reve | [Random event](/D/reve) | JoramSoch | 2020-11-19 |
+| D111 | cvlme | [Cross-validated log model evidence](/D/cvlme) | JoramSoch | 2020-11-19 |
+| D112 | ind-cond | [Conditional independence](/D/ind-cond) | JoramSoch | 2020-11-19 |
diff --git a/I/Definition_by_Topic.md b/I/Definition_by_Topic.md
@@ -27,6 +27,7 @@ title: "Definition by Topic"
 - [Coefficient of determination](/D/rsq)
 - [Conditional differential entropy](/D/dent-cond)
 - [Conditional entropy](/D/ent-cond)
+- [Conditional independence](/D/ind-cond)
 - [Conditional probability distribution](/D/dist-cond)
 - [Constant](/D/const)
 - [Continuous uniform distribution](/D/cuni)
@@ -35,6 +36,7 @@ title: "Definition by Topic"
 - [Covariance](/D/cov)
 - [Covariance matrix](/D/covmat)
 - [Cross-entropy](/D/ent-cross)
+- [Cross-validated log model evidence](/D/cvlme)
 - [Cumulant-generating function](/D/cgf)
 - [Cumulative distribution function](/D/cdf)
 
@@ -139,6 +141,8 @@ title: "Definition by Topic"
 
 ### R
 
+- [Random event](/D/reve)
+- [Random experiment](/D/rexp)
 - [Random matrix](/D/rmat)
 - [Random variable](/D/rvar)
 - [Random vector](/D/rvec)
diff --git a/I/Proof_by_Author.md b/I/Proof_by_Author.md
@@ -4,7 +4,7 @@ title: "Proof by Author"
 ---
 
 
-### JoramSoch (183 proofs)
+### JoramSoch (186 proofs)
 
 - [(Non-)Multiplicativity of the expected value](/P/mean-mult)
 - [Additivity of the Kullback-Leibler divergence for independent distributions](/P/kl-add)
@@ -29,6 +29,7 @@ title: "Proof by Author"
 - [Cumulative distribution function in terms of probability mass function of a discrete random variable](/P/cdf-pmf)
 - [Cumulative distribution function of a strictly decreasing function of a random variable](/P/cdf-sdfct)
 - [Cumulative distribution function of a strictly increasing function of a random variable](/P/cdf-sifct)
+- [Cumulative distribution function of the beta distribution](/P/beta-cdf)
 - [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)
@@ -60,6 +61,7 @@ title: "Proof by Author"
 - [Joint likelihood is the product of likelihood function and prior density](/P/jl-lfnprior)
 - [Kullback-Leibler divergence for the gamma distribution](/P/gam-kl)
 - [Kullback-Leibler divergence for the multivariate normal distribution](/P/mvn-kl)
+- [Kullback-Leibler divergence for the normal distribution](/P/norm-kl)
 - [Kullback-Leibler divergence for the normal-gamma distribution](/P/ng-kl)
 - [Law of the unconscious statistician](/P/mean-lotus)
 - [Linear transformation theorem for the matrix-normal distribution](/P/matn-ltt)
@@ -156,6 +158,7 @@ title: "Proof by Author"
 - [Quantile function of the continuous uniform distribution](/P/cuni-qf)
 - [Quantile function of the discrete uniform distribution](/P/duni-qf)
 - [Quantile function of the exponential distribution](/P/exp-qf)
+- [Quantile function of the gamma distribution](/P/gam-qf)
 - [Quantile function of the normal distribution](/P/norm-qf)
 - [Relation between gamma distribution and standard gamma distribution](/P/gam-sgam)
 - [Relation between gamma distribution and standard gamma distribution](/P/gam-sgam2)
diff --git a/I/Proof_by_Number.md b/I/Proof_by_Number.md
@@ -199,3 +199,6 @@ title: "Proof by Number"
 | P190 | cdf-pdf | [Cumulative distribution function in terms of probability density function of a continuous random variable](/P/cdf-pdf) | JoramSoch | 2020-11-12 |
 | P191 | pdf-cdf | [Probability density function is first derivative of cumulative distribution function](/P/pdf-cdf) | JoramSoch | 2020-11-12 |
 | P192 | qf-cdf | [Quantile function is inverse of strictly monotonically increasing cumulative distribution function](/P/qf-cdf) | JoramSoch | 2020-11-12 |
+| P193 | norm-kl | [Kullback-Leibler divergence for the normal distribution](/P/norm-kl) | JoramSoch | 2020-11-19 |
+| P194 | gam-qf | [Quantile function of the gamma distribution](/P/gam-qf) | JoramSoch | 2020-11-19 |
+| P195 | beta-cdf | [Cumulative distribution function of the beta distribution](/P/beta-cdf) | JoramSoch | 2020-11-19 |
diff --git a/I/Proof_by_Topic.md b/I/Proof_by_Topic.md
@@ -39,6 +39,7 @@ title: "Proof by Topic"
 - [Cumulative distribution function in terms of probability mass function of a discrete random variable](/P/cdf-pmf)
 - [Cumulative distribution function of a strictly decreasing function of a random variable](/P/cdf-sdfct)
 - [Cumulative distribution function of a strictly increasing function of a random variable](/P/cdf-sifct)
+- [Cumulative distribution function of the beta distribution](/P/beta-cdf)
 - [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)
@@ -92,6 +93,7 @@ title: "Proof by Topic"
 
 - [Kullback-Leibler divergence for the gamma distribution](/P/gam-kl)
 - [Kullback-Leibler divergence for the multivariate normal distribution](/P/mvn-kl)
+- [Kullback-Leibler divergence for the normal distribution](/P/norm-kl)
 - [Kullback-Leibler divergence for the normal-gamma distribution](/P/ng-kl)
 
 ### L
@@ -211,6 +213,7 @@ title: "Proof by Topic"
 - [Quantile function of the continuous uniform distribution](/P/cuni-qf)
 - [Quantile function of the discrete uniform distribution](/P/duni-qf)
 - [Quantile function of the exponential distribution](/P/exp-qf)
+- [Quantile function of the gamma distribution](/P/gam-qf)
 - [Quantile function of the normal distribution](/P/norm-qf)
 
 ### R
diff --git a/I/Proofs_without_Source.md b/I/Proofs_without_Source.md
@@ -19,6 +19,7 @@ title: "Proofs without Source"
 - [Exponential distribution is a special case of gamma distribution](/P/exp-gam)
 - [First raw moment is mean](/P/momraw-1st)
 - [Joint likelihood is the product of likelihood function and prior density](/P/jl-lfnprior)
+- [Kullback-Leibler divergence for the normal distribution](/P/norm-kl)
 - [Linear transformation theorem for the matrix-normal distribution](/P/matn-ltt)
 - [Log model evidence for multinomial observations](/P/mult-lme)
 - [Log model evidence for multivariate Bayesian linear regression](/P/mblr-lme)
diff --git a/I/Table_of_Contents.md b/I/Table_of_Contents.md
@@ -56,11 +56,11 @@ title: "Table of Contents"
    &emsp;&ensp; 1.4.12. **[Cumulative distribution function of continuous random variable](/P/cdf-pdf)** <br>
    &emsp;&ensp; 1.4.13. *[Quantile function](/D/qf)* <br>
    &emsp;&ensp; 1.4.14. **[Quantile function in terms of cumulative distribution function](/P/qf-cdf)** <br>
-   &emsp;&ensp; 1.4.11. *[Moment-generating function](/D/mgf)* <br>
-   &emsp;&ensp; 1.4.12. **[Moment-generating function of linear transformation](/P/mgf-ltt)** <br>
-   &emsp;&ensp; 1.4.13. **[Moment-generating function of linear combination](/P/mgf-lincomb)** <br>
-   &emsp;&ensp; 1.4.14. *[Cumulant-generating function](/D/cgf)* <br>
-   &emsp;&ensp; 1.4.15. *[Probability-generating function](/D/pgf)* <br>
+   &emsp;&ensp; 1.4.15. *[Moment-generating function](/D/mgf)* <br>
+   &emsp;&ensp; 1.4.16. **[Moment-generating function of linear transformation](/P/mgf-ltt)** <br>
+   &emsp;&ensp; 1.4.17. **[Moment-generating function of linear combination](/P/mgf-lincomb)** <br>
+   &emsp;&ensp; 1.4.18. *[Cumulant-generating function](/D/cgf)* <br>
+   &emsp;&ensp; 1.4.19. *[Probability-generating function](/D/pgf)* <br>
    
    1.5. Expected value <br>
    &emsp;&ensp; 1.5.1. *[Definition](/D/mean)* <br>
@@ -479,17 +479,17 @@ title: "Table of Contents"
    &emsp;&ensp; 3.2.2. **[Derivation](/P/lfe-der)** <br>
    &emsp;&ensp; 3.2.3. **[Calculation from log model evidences](/P/lfe-lme)** <br>
    
-   3.3. Bayes factor <br>
-   &emsp;&ensp; 3.3.1. *[Definition](/D/bf)* <br>
-   &emsp;&ensp; 3.3.2. **[Transitivity](/P/bf-trans)** <br>
-   &emsp;&ensp; 3.3.3. **[Computation using Savage-Dickey Density Ratio](/P/bf-sddr)** <br>
-   &emsp;&ensp; 3.3.4. **[Computation using Encompassing Prior Method](/P/bf-ep)** <br>
-   &emsp;&ensp; 3.3.5. *[Encompassing model](/D/encm)* <br>
-   
-   3.4. Log Bayes factor <br>
-   &emsp;&ensp; 3.4.1. *[Definition](/D/lbf)* <br>
-   &emsp;&ensp; 3.4.2. **[Derivation](/P/lbf-der)** <br>
-   &emsp;&ensp; 3.4.3. **[Calculation from log model evidences](/P/lbf-lme)** <br>
+   3.3. Log Bayes factor <br>
+   &emsp;&ensp; 3.3.1. *[Definition](/D/lbf)* <br>
+   &emsp;&ensp; 3.3.2. **[Derivation](/P/lbf-der)** <br>
+   &emsp;&ensp; 3.3.3. **[Calculation from log model evidences](/P/lbf-lme)** <br>
+   
+   3.4. Bayes factor <br>
+   &emsp;&ensp; 3.4.1. *[Definition](/D/bf)* <br>
+   &emsp;&ensp; 3.4.2. **[Transitivity](/P/bf-trans)** <br>
+   &emsp;&ensp; 3.4.3. **[Computation using Savage-Dickey Density Ratio](/P/bf-sddr)** <br>
+   &emsp;&ensp; 3.4.4. **[Computation using Encompassing Prior Method](/P/bf-ep)** <br>
+   &emsp;&ensp; 3.4.5. *[Encompassing model](/D/encm)* <br>
    
    3.5. Posterior model probability <br>
    &emsp;&ensp; 3.5.1. *[Definition](/D/pmp)* <br>
diff --git a/P/beta-cdf.md b/P/beta-cdf.md
@@ -54,7 +54,7 @@ $$ \label{eq:beta-cdf-app}
 \begin{split}
 F_X(x) &= \int_{0}^{x} \mathrm{Bet}(z; \alpha, \beta) \, \mathrm{d}z \\
 &= \int_{0}^{x} \frac{1}{\mathrm{B}(\alpha, \beta)} \, z^{\alpha-1} \, (1-z)^{\beta-1} \, \mathrm{d}z \\
-&= \frac{1}{B(x;a,b)} \int_{0}^{x} z^{\alpha-1} \, (1-z)^{\beta-1} \, \mathrm{d}z \; .
+&= \frac{1}{\mathrm{B}(\alpha, \beta)} \int_{0}^{x} z^{\alpha-1} \, (1-z)^{\beta-1} \, \mathrm{d}z \; .
 \end{split}
 $$
 
diff --git a/P/cuni-qf.md b/P/cuni-qf.md
@@ -57,7 +57,7 @@ $$ \label{eq:qf}
 Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .
 $$
 
-Thus, we have $Q_X(p) = -\infty$, if $p = 0$. When $p > 0$, it holds that
+Thus, we have $Q_X(p) = -\infty$, if $p = 0$. When $p > 0$, [it holds that](/P/qf-cdf)
 
 $$ \label{eq:exp-qf-s1}
 Q_X(p) = F_X^{-1}(x) \; .
diff --git a/P/exp-qf.md b/P/exp-qf.md
@@ -56,7 +56,7 @@ $$ \label{eq:qf}
 Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .
 $$
 
-Thus, we have $Q_X(p) = -\infty$, if $p = 0$. When $p > 0$, it holds that
+Thus, we have $Q_X(p) = -\infty$, if $p = 0$. When $p > 0$, [it holds that](/P/qf-cdf)
 
 $$ \label{eq:exp-qf-s1}
 Q_X(p) = F_X^{-1}(x) \; .
diff --git a/P/gam-cdf.md b/P/gam-cdf.md
@@ -70,7 +70,7 @@ $$
 
 With the definition of the lower incomplete gamma function
 
-$$ \label{eq:li-gam-fct}
+$$ \label{eq:low-inc-gam-fct}
 \gamma(s,x) = \int_{0}^{x} t^{s-1} \exp[-t] \, \mathrm{d}t \; ,
 $$
 
diff --git a/P/gam-kl.md b/P/gam-kl.md
@@ -26,12 +26,12 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let $x$ be a [random variable](/D/rvar). Assume two [gamma distributions](/D/gam) $P$ and $Q$ specifying the probability distribution of $x$ as
+**Theorem:** Let $X$ be a [random variable](/D/rvar). Assume two [gamma distributions](/D/gam) $P$ and $Q$ specifying the probability distribution of $X$ as
 
 $$ \label{eq:gams}
 \begin{split}
-P: \; x &\sim \mathrm{Gam}(a_1, b_1) \\
-Q: \; x &\sim \mathrm{Gam}(a_2, b_2) \; . \\
+P: \; X &\sim \mathrm{Gam}(a_1, b_1) \\
+Q: \; X &\sim \mathrm{Gam}(a_2, b_2) \; . \\
 \end{split}
 $$
 
diff --git a/P/lme-anc.md b/P/lme-anc.md
@@ -41,10 +41,10 @@ $$ \label{eq:LME}
 \mathrm{LME}(m) = \mathrm{Acc}(m) - \mathrm{Com}(m)
 $$
 
-where the accuracy term is the posterior expectation of the log-[likelihood function](/D/lf)
+where the accuracy term is the [posterior](/D/post) [expectation](/D/mean-lotus) of the [log-likelihood function](/D/llf)
 
 $$ \label{eq:Acc}
-\mathrm{Acc}(m) = \left\langle p(y|\theta,m) \right\rangle_{p(\theta|y,m)}
+\mathrm{Acc}(m) = \left\langle \log p(y|\theta,m) \right\rangle_{p(\theta|y,m)}
 $$
 
 and the complexity penalty is the [Kullback-Leibler divergence](/D/kl) of [posterior](/D/post) from [prior](/D/prior)
@@ -54,7 +54,7 @@ $$ \label{eq:Com}
 $$
 
 
-**Proof:** We consider Bayesian inference on data $y$ using model $m$ with parameters $\theta$. Then, [Bayes' theorem](/P/bayes-th) makes a statement about the posterior distribution, i.e. the probability of parameters, given the data and the model:
+**Proof:** We consider Bayesian inference on [data](/D/data) $y$ using [model](/D/gm) $m$ with parameters $\theta$. Then, [Bayes' theorem](/P/bayes-th) makes a statement about the [posterior distribution](/D/post), i.e. the probability of parameters, given the data and the model:
 
 $$ \label{eq:AnC-s1}
 p(\theta|y,m) = \frac{p(y|\theta,m) \, p(\theta|m)}{p(y|m)} \; .
@@ -81,7 +81,7 @@ $$
 By definition, the left-hand side is the log model evidence and the terms on the right-hand side correspond to the posterior expectation of the log-likelihood function and the Kullback-Leibler divergence of posterior from prior
 
 $$ \label{eq:LME-AnC}
-\mathrm{LME}(m) = \left\langle p(y|\theta,m) \right\rangle_{p(\theta|y,m)} - \mathrm{KL} \left[ p(\theta|y,m) \, || \, p(\theta|m) \right]
+\mathrm{LME}(m) = \left\langle \log p(y|\theta,m) \right\rangle_{p(\theta|y,m)} - \mathrm{KL} \left[ p(\theta|y,m) \, || \, p(\theta|m) \right]
 $$
 
 which proofs the partition given by \eqref{eq:LME}.
diff --git a/P/norm-kl.md b/P/norm-kl.md
@@ -25,8 +25,8 @@ username: "JoramSoch"
 
 $$ \label{eq:norms}
 \begin{split}
-P: \; X &\sim \mathrm{Gam}(\mu_1, \sigma_1^2) \\
-Q: \; X &\sim \mathrm{Gam}(\mu_2, \sigma_2^2) \; . \\
+P: \; X &\sim \mathcal{N}(\mu_1, \sigma_1^2) \\
+Q: \; X &\sim \mathcal{N}(\mu_2, \sigma_2^2) \; . \\
 \end{split}
 $$
 
diff --git a/S/Statistical_Models.md b/S/Statistical_Models.md
@@ -12,7 +12,7 @@ title: "Special: Statistical Models"
 | Multiple linear regression | *[mlr](/D/mlr)* | **[mlr-ols](/P/mlr-ols)**<br>**[mlr-wls](/P/mlr-wls)** | **[mlr-mle](/P/mlr-mle)** |  |  |  | **[mlr-pss](/P/mlr-pss)**<br>**[mlr-mat](/P/mlr-mat)**<br>**[mlr-idem](/P/mlr-idem)** |
 | Bayesian linear regression |  |  |  | **[blr-prior](/P/blr-prior)** | **[blr-post](/P/blr-post)** | **[blr-lme](/P/blr-lme)** | **[blr-pp](/P/blr-pp)**<br>**[blr-pcr](/P/blr-pcr)** |
 | **Multivariate<br>normal data** |  |  |  |  |  |  |  |
-| General linear model | *[glm](/D/mlr)* | **[glm-ols](/P/glm-ols)**<br>**[glm-wls](/P/glm-wls)** | **[glm-mle](/P/glm-mle)** |  |  |  |  |
+| General linear model | *[glm](/D/glm)* | **[glm-ols](/P/glm-ols)**<br>**[glm-wls](/P/glm-wls)** | **[glm-mle](/P/glm-mle)** |  |  |  |  |
 | Multivariate Bayesian linear regression |  |  |  | **[mblr-prior](/P/mblr-prior)** | **[mblr-post](/P/mblr-post)** | **[mblr-lme](/P/mblr-lme)** |  |
 | **Poisson data** |  |  |  |  |  |  |  |
 | Poisson-distributed data | *[poiss](/D/poiss-data)* |  | **[poiss-mle](/P/poiss-mle)** |  |  |  |  |

Original file line number	Diff line number	Diff line change
`@@ -33,4 +33,4 @@ username: "JoramSoch"`
`33`	`33`
`34`	`34`	`* formally, as a [measurable function](/D/meas-fct) $X$ defined on a [probability space](/D/prob-spc) $(\Omega, \mathcal{F}, P)$ that maps from a sample space $\Omega$ to the real numbers $\mathbb{R}$ using an event space $\mathcal{F}$ and a [probability function](/D/pmf) $P$;`
`35`	`35`
`36`		`-* more broadly, as any random quantity $X$ such as a [random scalar](/D/rvar), a [random vector](/D/rvec) or a [random matrix](/D/rmat).`
	`36`	`+* more broadly, as any random quantity $X$ such as a [random event](/D/reve), a [random scalar](/D/rvar), a [random vector](/D/rvec) or a [random matrix](/D/rmat).`