From f091bf0922c26238d16bf175a8ea916a16d11fba Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sat, 6 Nov 2021 21:47:08 +0100
Subject: Expand knowledge base

---
 content/know/concept/random-variable/index.pdc | 22 +++++++++++-----------
 1 file changed, 11 insertions(+), 11 deletions(-)

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

diff --git a/content/know/concept/random-variable/index.pdc b/content/know/concept/random-variable/index.pdc
index fe50b60..2a8643e 100644
--- a/content/know/concept/random-variable/index.pdc
+++ b/content/know/concept/random-variable/index.pdc
@@ -119,27 +119,27 @@ $$\begin{aligned}
 
 ## Expectation value
 
-The **expectation value** $\mathbf{E}(X)$ of a random variable $X$
+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).
 For continuous and discrete sample spaces $\Omega$, respectively:
 
 $$\begin{aligned}
-    \mathbf{E}(X)
+    \mathbf{E}[X]
     = \int_{-\infty}^\infty x \: f_X(x) \dd{x}
     \qquad \mathrm{or} \qquad
-    \mathbf{E}(X)
+    \mathbf{E}[X]
     = \sum_{i = 1}^N x_i \: P(X \!=\! x_i)
 \end{aligned}$$
 
 However, $f_X(x)$ is not guaranteed to exist,
 and the distinction between continuous and discrete is cumbersome.
-A more general definition of $\mathbf{E}(X)$
+A more general definition of $\mathbf{E}[X]$
 is the following Lebesgue-Stieltjes integral,
 since $F_X(x)$ always exists:
 
 $$\begin{aligned}
-    \mathbf{E}(X)
+    \mathbf{E}[X]
     = \int_{-\infty}^\infty x \dd{F_X(x)}
 \end{aligned}$$
 
@@ -147,25 +147,25 @@ This is valid for any sample space $\Omega$.
 Or, equivalently, a Lebesgue integral can be used:
 
 $$\begin{aligned}
-    \mathbf{E}(X)
+    \mathbf{E}[X]
     = \int_\Omega X(\omega) \dd{P(\omega)}
 \end{aligned}$$
 
 An expectation value defined in this way has many useful properties,
 most notably linearity.
 
-We can also define the familiar **variance** $\mathbf{V}(X)$
+We can also define the familiar **variance** $\mathbf{V}[X]$
 of a random variable $X$ as follows:
 
 $$\begin{aligned}
-    \mathbf{V}(X)
-    = \mathbf{E}\big( (X - \mathbf{E}(X))^2 \big)
-    = \mathbf{E}(X^2) - \big(\mathbf{E}(X)\big)^2
+    \mathbf{V}[X]
+    = \mathbf{E}\big[ (X - \mathbf{E}[X])^2 \big]
+    = \mathbf{E}[X^2] - \big(\mathbf{E}[X]\big)^2
 \end{aligned}$$
 
 
 
 ## References
-1.  U.F. Thygesen,
+1.  U.H. Thygesen,
     *Lecture notes on diffusions and stochastic differential equations*,
     2021, Polyteknisk Kompendie.
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