From 1d700ab734aa9b6711eb31796beb25cb7659d8e0 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Tue, 20 Dec 2022 20:11:25 +0100
Subject: More improvements to knowledge base

---
 source/know/concept/random-variable/index.md | 17 ++++++++++-------
 1 file changed, 10 insertions(+), 7 deletions(-)

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

diff --git a/source/know/concept/random-variable/index.md b/source/know/concept/random-variable/index.md
index ecb8e96..a6cbc8b 100644
--- a/source/know/concept/random-variable/index.md
+++ b/source/know/concept/random-variable/index.md
@@ -17,6 +17,7 @@ Here, we will describe the formal mathematical definition
 of a random variable.
 
 
+
 ## Probability space
 
 A **probability space** or **probability triple** $$(\Omega, \mathcal{F}, P)$$
@@ -24,7 +25,7 @@ is the formal mathematical model of a given **stochastic experiment**,
 i.e. a process with a random outcome.
 
 The **sample space** $$\Omega$$ is the set
-of all possible outcomes $$\omega$$ of the experimement.
+of all possible outcomes $$\omega$$ of the stochastic experiment.
 Those $$\omega$$ are selected randomly according to certain criteria.
 A subset $$A \subset \Omega$$ is called an **event**,
 and can be regarded as a true statement about all $$\omega$$ in that $$A$$.
@@ -34,7 +35,7 @@ that are interesting to us,
 i.e. we have subjectively chosen $$\mathcal{F}$$
 based on the problem at hand.
 Since events $$A$$ represent statements about outcomes $$\omega$$,
-and we would like to use logic on those statemenets,
+and we would like to use logic on those statements,
 we demand that $$\mathcal{F}$$ is a [$$\sigma$$-algebra](/know/concept/sigma-algebra/).
 
 Finally, the **probability measure** or **probability function** $$P$$
@@ -52,6 +53,7 @@ if $$\Omega$$ is continuous, all $$\omega$$ have zero probability,
 while intervals $$A$$ can have nonzero probability.
 
 
+
 ## Random variable
 
 Once we have a probability space $$(\Omega, \mathcal{F}, P)$$,
@@ -91,7 +93,7 @@ $$X^{-1}$$ 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 often stated as "$$X$$ is *$$\mathcal{F}$$-measurable*".
 
 Related to $$\mathcal{F}$$ is the **information**
 obtained by observing a random variable $$X$$.
@@ -111,7 +113,7 @@ 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"*
+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 $$Y$$.
@@ -145,11 +147,12 @@ $$\begin{aligned}
 \end{aligned}$$
 
 
+
 ## Expectation value
 
 The **expectation value** $$\mathbf{E}[X]$$ of a random variable $$X$$
 can be defined in the familiar way, as the sum/integral
-of every possible value of $$X$$ mutliplied by the corresponding probability (density).
+of every possible value of $$X$$ multiplied by the corresponding probability (density).
 For continuous and discrete sample spaces $$\Omega$$, respectively:
 
 $$\begin{aligned}
@@ -163,7 +166,7 @@ $$\begin{aligned}
 However, $$f_X(x)$$ is not guaranteed to exist,
 and the distinction between continuous and discrete is cumbersome.
 A more general definition of $$\mathbf{E}[X]$$
-is the following Lebesgue-Stieltjes integral,
+is the following *Lebesgue-Stieltjes integral*,
 since $$F_X(x)$$ always exists:
 
 $$\begin{aligned}
@@ -172,7 +175,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 This is valid for any sample space $$\Omega$$.
-Or, equivalently, a Lebesgue integral can be used:
+Or, equivalently, a *Lebesgue integral* can be used:
 
 $$\begin{aligned}
     \mathbf{E}[X]
-- 
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