From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Sat, 24 Aug 2019 04:33:48 +0000 (+1000)
Subject: [methods] fill gaps in statistics notes
X-Git-Tag: yr12~57
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/5753a4ea89655a60374b4bc423bc2cd461bf28c1

[methods] fill gaps in statistics notes
---

diff --git a/methods/statistics.pdf b/methods/statistics.pdf
index 0929ba0..7a81019 100644
Binary files a/methods/statistics.pdf and b/methods/statistics.pdf differ
diff --git a/methods/statistics.tex b/methods/statistics.tex
index 8ac24bf..1ad360e 100644
--- a/methods/statistics.tex
+++ b/methods/statistics.tex
@@ -69,7 +69,8 @@
       \begin{align*}
         \sigma^2=\operatorname{Var}(x) &= \sum_{i=1}^n p_i (x_i-\mu)^2 \\
         &= \sum (x-\mu)^2 \times \Pr(X=x) \\
-        &= \sum x^2 \times p(x) - \mu^2
+        &= \sum x^2 \times p(x) - \mu^2 \\
+        &= \operatorname{E}(X^2) - [\operatorname{E}(X)]^2
       \end{align*}
     \item \textbf{Standard deviation $\sigma$} - measure of spread in the original magnitude of the data. Found by taking square root of the variance:
       \begin{align*}
@@ -90,6 +91,9 @@
     E(X+Y) &= E(X) + E(Y) \tag{for two random variables}
   \end{align*}
 
+  \subsubsection*{Variance theorems}
+
+  \[ \operatorname{Var}(aX \pm bY \pm c) = a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y) \]
 
   \section{Binomial Theorem}