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zAq+>hM@M*7({OH$0_Q8!CFAS0xS7^k_(4jhge}KgT^`?-@U=fHxVEjV9InXa8_j_4 zEy^=CmzddCBF-DW3embSo7cSdd#su=W5NUDtcY)wlKIyqkC{ zQ*h3%$b7ab9t<<*VX0&_nv@P#s8bshTHo8{57)?>TqTHYs4GXecW$-crF+C#ZP!TZ zOs&c@{3|Cnv$hM4-s=^a0)7$K1fqw8Vh2a_yd{5z3m1Pe&)qM-K1EVNCbJ|>8j!|+ zybS_CP+*Lz{*QW`hb0YX35*25st?DcYVB=K%)<)Dq)DvHLd-_Y@;#^G=;%hw%|pyg ztOCa*<6z-P%)|WO194(qehzVNNl{iFb}>;lW==5{adt^oZf+K477kJxn4Ctsd&O6+>Q=$Ta;birfvm~9AUedL)pMYifQHR>!>U27 zdtoMW5=g%}WN>B}vG_1!oIvHVKAf1r5PHD}B5m!l?g_{5X 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} -- 2.47.1