[methods] fill gaps in statistics notes

diff --git a/methods/statistics.pdf b/methods/statistics.pdf
index 0929ba09c6bc5f4402728543f81cac3f1a88f066..7a81019f20b07eedb87072e084bf7bf8305bb088 100644 (file)
Binary files a/methods/statistics.pdf and b/methods/statistics.pdf differ

diff --git a/methods/statistics.tex b/methods/statistics.tex
index 8ac24bff039f283e48ad2349b3bcd2cdae081211..1ad360ef561d47af2697382a0582d23f59f877fa 100644 (file)
--- 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) \\
       \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*}
       \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*}
 
     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}
 
 
   \section{Binomial Theorem}