From d65c435cc2937dad127d8f12242fc327e295fa62 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sun, 11 Aug 2019 14:45:48 +1000 Subject: [PATCH 1/1] [methods] clean up statistics notes --- methods/statistics.pdf | Bin 171504 -> 182990 bytes methods/statistics.tex | 128 ++++++++++++++++++++++------------------- 2 files changed, 68 insertions(+), 60 deletions(-) diff --git a/methods/statistics.pdf b/methods/statistics.pdf index bccb7d23b36684c039b60729dde560a93c1c6a95..0929ba09c6bc5f4402728543f81cac3f1a88f066 100644 GIT binary patch delta 119504 zcmV)AK*YcBy9&;g3y>lNGc`0hm+_4PD1Y5sTXW;K6@K@xP~WI&gy4SJOggihCY{Y> z({1Bs(r)rVOSHvSA{~-i+cW*~J-F~H=wipa-S)+#01!Ahm+xEvhj|A zZ}iL{BdMHSUeAO?(y*B(44$*g)$CpJw<2%m3&vQ&?1Lw7vSwY*7gQyy+PD1M_kaJ; zq)lEo`4XCyQ7U;^tmY)CviX7}cYe%wWtDHUYW~aRx1WE*$Y5z^1d>_+U2a(dA7PA> zw||{eo&5RL%U7?zdwuzL`{BjcmlwZY5EzMPq$3%_RG80}+lzO<;MofLzMWyjjGo=O zk=q&91Ze}Wo7oQ+f4RVc33$j1B7bLuBgQ2iutO<?_!UafT=u%>rmLJSg~^-KNQJH~A8*xhz0SHE*>vqIX0}WDI|9#e240 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80xI`k55vhM7dJ`Fvr&W39t4yP#zSf z`}a$W|Mz9_gAglg9Lq-Z72xh<}t}xj*X|f=TAz65ZpJ)Xd zLEL3$-YzR5pA}Ln${K%t_jse=w+Nt-S8KUR_{H*c&*iWr@wBdcdYt)-{OJMUF#jqP zSGp*PtNHa~k^k(!QQ-HbXYN9xgN#opF#*T0|L>Y+x?+3wuFx_9>E!+s*z1^&B TFt`P||Mm750D!!@0>*y dx \quad \text{(for pdf } f \text{)} - &= \sum_{-\infty}^{\infty} + \overline{x} = \mu = E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f} \tag{where \(f =\) absolute frequency} \\ + &= \sum_{i=1}^n \left[ x_i \cdot \Pr(X=x_i) \right] \tag{for \(n\) values of \(x\)}\\ + &= \int_{-\infty}^{\infty} (x\cdot f(x)) \> dx \tag{for pdf \(f\)} \end{align*} \item \textbf{Mode} - most popular value (has highest probability of \(X\) values). Multiple modes can exist if \(>1 \> X\) value have equal-highest probability. Number must exist in distribution. - \item \textbf{Median \(m\)} - the value of \(x\) such that \(\Pr(X \le m) = \Pr(X \ge m) = 0.5\). If \(m > 0.5\), then value of \(X\) that is reached is the median of \(X\). If \(m = 0.5 = 0.5\), then \(m\) is halfway between this value and the next. + \item \textbf{Median \(m\)} - the value of \(x\) such that \(\Pr(X \le m) = \Pr(X \ge m) = 0.5\). If \(m > 0.5\), then value of \(X\) that is reached is the median of \(X\). If \(m = 0.5 = 0.5\), then \(m\) is halfway between this value and the next. To find \(m\), add values of \(X\) from smallest to alrgest until the sum reaches 0.5. \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \] \item \textbf{Variance $\sigma^2$} - measure of spread of data around the mean. Not the same magnitude as the original data. For distribution \(x_1 \mapsto p_1, x_2 \mapsto p_2, \dots, x_n \mapsto p_n\): \begin{align*} @@ -72,17 +71,23 @@ &= \sum (x-\mu)^2 \times \Pr(X=x) \\ &= \sum x^2 \times p(x) - \mu^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: $\sigma =\operatorname{sd}(X)=\sqrt{\operatorname{Var}(X)}$ + \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*} + \sigma &= \operatorname{sd}(X) \\ + &= \sqrt{\operatorname{Var}(X)} + \end{align*} \end{itemize} - \subsubsection{Expectation theorems} + \subsubsection*{Expectation theorems} + + For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\). \begin{align*} - E(aX \pm b) &= aE(X) \pm b \\ - E(z) &= z \\ - E(X+Y) &= E(X) + E(Y) \\ - E(X)^n &= \Sigma x^n \cdot p(x) \\ - &\ne [E(X)]^2 + E(X^n) &= \Sigma x^n \cdot p(x) \tag{non-linear function} \\ + &\ne [E(X)]^n \\ + E(aX \pm b) &= aE(X) \pm b \tag{linear function} \\ + E(b) &= b \tag{for constant \(b \in \mathbb{R}\)}\\ + E(X+Y) &= E(X) + E(Y) \tag{for two random variables} \end{align*} @@ -94,6 +99,7 @@ &= \sum_{k=0}^n {n \choose k} x^k y^{n-k} \end{align*} + \subsubsection*{Patterns} \begin{enumerate} \item powers of \(x\) decrease \(n \rightarrow 0\) \item powers of \(y\) increase \(0 \rightarrow n\) @@ -101,14 +107,19 @@ \item Number of terms in \((x+a)^n\) expanded \& simplified is \(n+1\) \end{enumerate} - Combinations: \(^n\text{C}_r = {N\choose k}\) (binomial coefficient) + \subsubsection*{Combinatorics} + + \[ \text{Binomial coefficient:} \quad ^n\text{C}_r = {N\choose k} \] + \begin{itemize} \item Arrangements \({n \choose k} = \frac{n!}{(n-r)}\) \item Combinations \({n \choose k} = \frac{n!}{r!(n-r)!}\) \item Note \({n \choose k} = {n \choose k-1}\) \end{itemize} - \subsubsection{Pascal's Triangle} + \colorbox{cas}{On CAS:} (soft keyboard) \keystroke{\(\downarrow\)} \(\rightarrow\) \keystroke{Advanced} \(\rightarrow\) \verb;nCr(n,cr); + + \subsubsection*{Pascal's Triangle} \begin{tabular}{>{$}l<{$\hspace{12pt}}*{13}{c}} n=\cr0&&&&&&&1&&&&&&\\ @@ -120,50 +131,47 @@ 6&1&&6&&15&&20&&15&&6&&1 \end{tabular} - \colorbox{cas}{On CAS:} (soft keys) \keystroke{\(\downarrow\)} \(\rightarrow\) \keystroke{Advanced} \(\rightarrow\) \verb;nCr(n,cr); - \section{Binomial distributions} (aka Bernoulli distributions) \begin{align*} - \Pr(X=x) &= {n \choose x} p^x (1-p)^{n-x} \\ + \text{Defined by} \quad X &\sim \operatorname{Bi}(n,p) \\ + \implies \Pr(X=x) &= {n \choose x} p^x (1-p)^{n-x} \\ &= {n \choose x} p^x q^{n-x} \end{align*} + where: + \begin{description} + \item \(n\) is the number of trials + \item There are two possible outcomes: \(S\) or \(F\) + \item \(\Pr(\text{success}) = p\) + \item \(\Pr(\text{failure}) = 1-p = q\) + \end{description} + + \subsection*{Conditions for a binomial variable/distribution} \begin{enumerate} \item Two possible outcomes: \textbf{success} or \textbf{failure} \item \(\Pr(\text{success})\) is constant across trials (also denoted \(p\)) \item Finite number \(n\) of independent trials \end{enumerate} - If these conditions are met, then it is a Binomial Random Variable. This variable is said to have a \textit{binomial probability distribution}. - - \begin{itemize} - \item \(n\) is the number of trials - \item There are two possible outcomes: \(S\) or \(F\) - \item \(\Pr(\text{success}) = p\) - \item \(\Pr(\text{failure}) = 1-p = q\) - \item Shorthand notation: \(X \sim \operatorname{Bi}(n,p)\) - \end{itemize} - - \colorbox{cas}{On CAS:} Main \(\rightarrow\) Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPDf; \\ - Input \verb;x; (no. of successes), \verb;numtrial; (no. of trials), \verb;pos; (probbability of success) - - \subsection{Applications of binomial distributions} - - \[ \Pr(X \ge a) = 1 - \Pr(X < a) \] - - \subsection{Expected value of a binomial distribution} + \subsection*{\colorbox{cas}{Solve on CAS}} + + Main \(\rightarrow\) Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPDf; - \[ E(X \sim \operatorname{Bi}(n,p))=np \] + \hspace{2em} Input \verb;x; (no. of successes), \verb;numtrial; (no. of trials), \verb;pos; (probbability of success) - \subsection{Variance} + \subsection*{Properties of \(X \sim \operatorname{Bi}(n,p)\)} - \[ \sigma^2(X) = np(1-p) \] + \begin{align*} + \textbf{Mean} \hspace{-4cm} &&\mu(X) &= np \\ + \textbf{Variance} \hspace{-4cm} &&\sigma^2(X) &= np(1-p) \\ + \textbf{s.d.} \hspace{-4cm} &&\sigma(X) &= \sqrt{np(1-p)} + \end{align*} - \subsection{Standard deviation} + \subsection*{Applications of binomial distributions} - \[ \sigma(X) = \sqrt{np(1-p)} \] + \[ \Pr(X \ge a) = 1 - \Pr(X < a) \] \end{document} -- 2.47.1