corrected some pages

Several small mistakes/errors were corrected in several proofs/definitions.

Author: JoramSoch

D/rexp.md

@@ -10,7 +10,7 @@ date: 2020-11-19 04:10:00
 title: "Random experiment"
 chapter: "General Theorems"
 section: "Probability theory"
-topic: "Random variables"
+topic: "Random experiments"
 definition: "Random experiment"
 
 sources:
@@ -29,10 +29,10 @@ username: "JoramSoch"
 
 **Definition:** A random experiment is any repeatable procedure that [results in one](/D/rvar) out of a well-defined set of possible outcomes.
 
-* The set of possible outcomes is called sample space.
+* The set of possible outcomes is called [sample space](/D/samp-spc).
 
 * A set of zero or more outcomes is called a [random event](/D/reve).
 
 * A function that maps from events to probabilities is called a [probability function](/D/dist).
 
-Together, sample space, event space and probability function characterize a random experiment.
+Together, [sample space](/D/samp-spc), [event space](/D/eve-spc) and [probability function](/D/prob-spc) characterize a random experiment.

D/rvar.md

@@ -31,6 +31,6 @@ username: "JoramSoch"
 
 * informally, as a real number $X \in \mathbb{R}$ whose value is the outcome of a [random experiment](/D/rexp);
 
-* 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/dist) $P$;
+* formally, as a [measurable function](/D/meas-fct) $X$ defined on a [probability space](/D/prob-spc) $(\Omega, \mathcal{E}, P)$ that maps from a [sample space](/D/samp-spc) $\Omega$ to the real numbers $\mathbb{R}$ using an [event space](/D/eve-spc) $\mathcal{E}$ and a [probability function](/D/dist) $P$;
 
 * 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).

I/ToC.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let $X$, $Y$ and $Z$ be [random variables](/D/rvar) defined on the same [probability space](/D/prob-spc) and assume that the [covariance](/D/cov) of $X$ and $Y$ is finite. Then, the sum of the [expectation](/D/mean) of the conditional covariance and the [covariance](/D/cov) of the conditional expectations of $X$ and $Y$ given $Z$ is equal to the [covariance](/D/cov) of $X$ and $Y$:
+**Theorem:** (law of total covariance, also called "conditional covariance formula") Let $X$, $Y$ and $Z$ be [random variables](/D/rvar) defined on the same [probability space](/D/prob-spc) and assume that the [covariance](/D/cov) of $X$ and $Y$ is finite. Then, the sum of the [expectation](/D/mean) of the conditional covariance and the [covariance](/D/cov) of the conditional expectations of $X$ and $Y$ given $Z$ is equal to the [covariance](/D/cov) of $X$ and $Y$:
 
 $$ \label{eq:cov-tot}
 \mathrm{Cov}(X,Y) = \mathrm{E}[\mathrm{Cov}(X,Y \vert Z)] + \mathrm{Cov}[\mathrm{E}(X \vert Z),\mathrm{E}(Y \vert Z)] \; .

P/cov-tot.md

@@ -42,12 +42,12 @@ $$ \label{eq:mean-lin}
 \end{split}
 $$
 
-for [random variables](/D/rvar) $X$ and $Y$ and a constant $a$.
+for [random variables](/D/rvar) $X$ and $Y$ and a [constant](/D/const) $a$.
 
 
 **Proof:**
 
-1) If $X$ and $Y$ are discrete random variables, the [expected value](/D/mean) is
+1) If $X$ and $Y$ are [discrete random variables](/D/rvar-disc), the [expected value](/D/mean) is
 
 $$ \label{eq:mean-disc}
 \mathrm{E}(X) = \sum_{x \in \mathcal{X}} x \cdot f_X(x)
@@ -82,7 +82,7 @@ $$ \label{eq:mean-lin-s2-disc}
 $$
 
 <br>
-2) If $X$ and $Y$ are continuous random variables, the [expected value](/D/mean) is
+2) If $X$ and $Y$ are [continuous random variables](/D/rvar-disc), the [expected value](/D/mean) is
 
 $$ \label{eq:mean-cont}
 \mathrm{E}(X) = \int_{\mathcal{X}} x \cdot f_X(x) \, \mathrm{d}x
@@ -117,4 +117,4 @@ $$ \label{eq:mean-lin-s2-cont}
 $$
 
 <br>
-Collectively, this shows that both requirements for linearity are fulfilled for the expected value, for discrete as well as for continuous random variables.
+Collectively, this shows that both requirements for linearity are fulfilled for the [expected value](/D/mean), for [discrete](/D/rvar-disc) as well as for [continuous](/D/rvar-disc) random variables.

P/mean-lin.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let $X$ be a [random variable](/D/rvar) with [expected value](/D/mean) $\mathrm{E}(X)$ and let $Y$ be any [random variable](/D/var) defined on the same [probability space](/D/prob-spc). Then, the [expected value](/D/mean) of the [conditional expectation](/D/mean-cond) of $X$ given $Y$ is the same as the [expected value](/D/mean) of $X$:
+**Theorem:** (law of total expectation, also called "law of iterated expectations") Let $X$ be a [random variable](/D/rvar) with [expected value](/D/mean) $\mathrm{E}(X)$ and let $Y$ be any [random variable](/D/var) defined on the same [probability space](/D/prob-spc). Then, the [expected value](/D/mean) of the [conditional expectation](/D/mean-cond) of $X$ given $Y$ is the same as the [expected value](/D/mean) of $X$:
 
 $$ \label{eq:mean-tot}
 \mathrm{E}(X) = \mathrm{E}[\mathrm{E}(X \vert Y)] \; .

P/mean-tot.md

@@ -33,7 +33,7 @@ where $1_n$ is an $n \times 1$ vector of ones, $x$ is the $n \times 1$ single pr
 **Proof:** Without loss of generality, consider the [simple linear regression case with uncorrelated errors](/D/slr):
 
 $$ \label{eq:slr}
-y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \; \varepsilon_i \sim \mathcal{N}(0, \sigma^2) \; .
+y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \; \varepsilon_i \sim \mathcal{N}(0, \sigma^2), \; i = 1,\ldots,n \; .
 $$
 
 In matrix notation and using the [multivariate normal distribution](/D/mvn), this can also be written as

P/slr-mlr.md

@@ -11,7 +11,7 @@ title: "Law of total variance"
 chapter: "General Theorems"
 section: "Probability theory"
 topic: "Variance"
-theorem: "Law of total expectation"
+theorem: "Law of total variance"
 
 sources:
   - authors: "Wikipedia"
@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Theorem:** Let $X$ and $Y$ be [random variables](/D/rvar) defined on the same [probability space](/D/prob-spc) and assume that the [variance](/D/var) of $Y$ is finite. Then, the sum of the [expectation](/D/mean) of the conditional variance and the [variance](/D/var) of the conditional expectation of $Y$ given $X$ is equal to the [variance](/D/var) of $Y$:
+**Theorem:** (law of total variance, also called "conditional variance formula") Let $X$ and $Y$ be [random variables](/D/rvar) defined on the same [probability space](/D/prob-spc) and assume that the [variance](/D/var) of $Y$ is finite. Then, the sum of the [expectation](/D/mean) of the conditional variance and the [variance](/D/var) of the conditional expectation of $Y$ given $X$ is equal to the [variance](/D/var) of $Y$:
 
 $$ \label{eq:var-tot}
 \mathrm{Var}(Y) = \mathrm{E}[\mathrm{Var}(Y \vert X)] + \mathrm{Var}[\mathrm{E}(Y \vert X)] \; .