From 62759ea3f910fae2617d033bf8f878d7574f4edd Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 7 Nov 2021 19:34:18 +0100
Subject: Expand knowledge base, reorganize measure theory, update gitignore

---
 content/know/concept/random-variable/index.pdc | 40 ++++++++++++++++++++++++--
 1 file changed, 38 insertions(+), 2 deletions(-)

(limited to 'content/know/concept/random-variable')

diff --git a/content/know/concept/random-variable/index.pdc b/content/know/concept/random-variable/index.pdc
index 2a8643e..bc41744 100644
--- a/content/know/concept/random-variable/index.pdc
+++ b/content/know/concept/random-variable/index.pdc
@@ -73,7 +73,8 @@ $$\begin{aligned}
     \quad \mathrm{for\:any\:} B \in \mathcal{B}(\mathbb{R}^n)
 \end{aligned}$$
 
-In other words, for a given Borel set (see $\sigma$-algebra) $B \in \mathcal{B}(\mathbb{R}^n)$,
+In other words, for a given Borel set
+(see [$\sigma$-algebra](/know/concept/sigma-algebra/)) $B \in \mathcal{B}(\mathbb{R}^n)$,
 the set of all outcomes $\omega \in \Omega$ that satisfy $X(\omega) \in B$
 must form a valid event; this set must be in $\mathcal{F}$.
 The point is that we need to be able to assign probabilities
@@ -94,7 +95,38 @@ $X^{-1}$ can be regarded as the inverse of $X$:
 it maps $B$ to the event for which $X \in B$.
 With this, our earlier requirement that $X$ be measurable
 can be written as: $X^{-1}(B) \in \mathcal{F}$ for any $B \in \mathcal{B}(\mathbb{R}^n)$.
-This is also often stated as *"$X$ is $\mathcal{F}$-measurable"*.
+This is also often stated as "$X$ is *$\mathcal{F}$-measurable"*.
+
+Related to $\mathcal{F}$ is the **information**
+obtained by observing a random variable $X$.
+Let $\sigma(X)$ be the information generated by observing $X$,
+i.e. the events whose occurrence can be deduced from the value of $X$,
+or, more formally:
+
+$$\begin{aligned}
+    \sigma(X)
+    = X^{-1}(\mathcal{B}(\mathbb{R}^n))
+    = \{ A \in \mathcal{F} : A = X^{-1}(B) \mathrm{\:for\:some\:} B \in \mathcal{B}(\mathbb{R}^n) \}
+\end{aligned}$$
+
+In other words, if the realized value of $X$ is
+found to be in a certain Borel set $B \in \mathcal{B}(\mathbb{R}^n)$,
+then the preimage $X^{-1}(B)$ (i.e. the event yielding this $B$)
+is known to have occurred.
+
+In general, given any $\sigma$-algebra $\mathcal{H}$,
+a variable $Y$ is said to be *"$\mathcal{H}$-measurable"*
+if $\sigma(Y) \subseteq \mathcal{H}$,
+so that $\mathcal{H}$ contains at least
+all information extractable from $Y$.
+
+Note that $\mathcal{H}$ can be generated by another random variable $X$,
+i.e. $\mathcal{H} = \sigma(X)$.
+In that case, the **Doob-Dynkin lemma** states
+that $Y$ is only $\sigma(X)$-measurable
+if $Y$ can always  be computed from $X$,
+i.e. there exists a function $f$ such that
+$Y(\omega) = f(X(\omega))$ for all $\omega \in \Omega$.
 
 Now, we are ready to define some familiar concepts from probability theory.
 The **cumulative distribution function** $F_X(x)$ is
@@ -163,6 +195,10 @@ $$\begin{aligned}
     = \mathbf{E}[X^2] - \big(\mathbf{E}[X]\big)^2
 \end{aligned}$$
 
+It is also possible to calculate expectation values and variances
+adjusted to some given event information:
+see [conditional expectation](/know/concept/conditional-expectation/).
+
 
 
 ## References
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