\documentclass[10pt,a4paper]{article}

% Packages
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%\usepackage{opensans}          % Can't make this work so far. Shame. Would be lovely.
\usepackage[normalem]{ulem}     % For underlining links
% Most of the following are not required for the majority
% of cheat sheets but are needed for some symbol support.
\usepackage{amsmath}            % Symbols
\usepackage{MnSymbol}           % Symbols
\usepackage{wasysym}            % Symbols
%\usepackage[english,german,french,spanish,italian]{babel}              % Languages

% Document Info
\author{t847222}
\pdfinfo{
  /Title (mat2377.pdf)
  /Creator (Cheatography)
  /Author (t847222)
  /Subject (MAT2377 Cheat Sheet)
}

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% Commands
\newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour
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% Header and Footer
\pagestyle{fancy}
\fancyhead{} % Set header to blank
\fancyfoot{} % Set footer to blank
\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
    \SetRowColor{DarkBackground}
    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
    }
\end{tabulary}
\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{MAT2377 Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{t847222} via \textcolor{DarkBackground}{\uline{cheatography.com/80028/cs/19345/}}}
\end{tabulary}
\end{multicols}}

\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}t847222 \\
  \uline{cheatography.com/t847222} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 13th April, 2019.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
  \SetRowColor{white}
  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
\raggedright
\raggedcolumns

% Set font size to small. Switch to any value
% from this page to resize cheat sheet text:
% www.emerson.emory.edu/services/latex/latex_169.html
\footnotesize % Small font.

\begin{multicols*}{4}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Classical + Relative}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{P(A) = N(A)/N(S) \newline % Row Count 1 (+ 1)
P(A) = f(A)/n% Row Count 2 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Conditional}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{P(A|B) = P(A∩B)/P(B)% Row Count 1 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{A given B}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{CDF}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{F(x) = P(X≤x) = Σf(x\_i)% Row Count 1 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Joint PMF}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{p(x,y) = P(X=x, Y=y) = P(\{X=x\}∩\{Y=y\})% Row Count 1 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Geometric Distribution}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{X = \# of trials until 1\textasciicircum{}st\textasciicircum{} success \newline % Row Count 1 (+ 1)
X \textasciitilde{} g(p) \newline % Row Count 2 (+ 1)
f(x) = (1-p)\textasciicircum{}x-1\textasciicircum{}p, for x=1,2,... \newline % Row Count 3 (+ 1)
F(x) = 1-(1-p)\textasciicircum{}x\textasciicircum{}, for x=1,2,... \newline % Row Count 4 (+ 1)
E{[}X{]} = 1/p \newline % Row Count 5 (+ 1)
V{[}X{]} = (1-p)/p\textasciicircum{}2\textasciicircum{}% Row Count 6 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Continuous Variable}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{P(a\textless{}X\textless{}b) = ∫f(x)dx = F(b)-F(a) \newline % Row Count 1 (+ 1)
f(x) = F'(x) \newline % Row Count 2 (+ 1)
F(x) = P(X\textless{}x) = ∫f(t)dt \newline % Row Count 3 (+ 1)
E{[}X{]} = ∫xf(x)dx \newline % Row Count 4 (+ 1)
V{[}X{]} = ∫x\textasciicircum{}2\textasciicircum{}f(x)dx-E{[}X{]}\textasciicircum{}2\textasciicircum{} \newline % Row Count 5 (+ 1)
E{[}g(X){]} = ∫g(x)f(x)dx \newline % Row Count 6 (+ 1)
V{[}g(X){]} = ∫(g(x))\textasciicircum{}2\textasciicircum{}f(x)dx=E{[}g(X){]}\textasciicircum{}2\textasciicircum{}% Row Count 7 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Normal Distribution}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{f(x) = 1/√(2πσ\textasciicircum{}2\textasciicircum{})*e\textasciicircum{}-(x-μ)\textasciicircum{}2/(2σ\textasciicircum{}2)\textasciicircum{}, -∞\textless{}x\textless{}∞ \newline % Row Count 2 (+ 2)
X \textasciitilde{} N(μ, σ\textasciicircum{}2\textasciicircum{}) \newline % Row Count 3 (+ 1)
E{[}X{]} = μ \newline % Row Count 4 (+ 1)
V{[}X{]} = σ\textasciicircum{}2\textasciicircum{}% Row Count 5 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Sample Mean}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{x̄ = Σx\_i/n% Row Count 1 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Box Plot}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{3.833cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/t847222_1555104876_mat.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{Describe histogram: skewness, uni/bimodal}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Constructing Confidence Interval}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{P = Y/n \newline % Row Count 1 (+ 1)
Y \textasciitilde{} b(n,p) \newline % Row Count 2 (+ 1)
Z = (P-p)/√(p(1-p)n) \textasciitilde{} N(0,1) \newline % Row Count 3 (+ 1)
E = z\_{[}α/2{]}√(p(1-p)/n)% Row Count 4 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Sample Correlation}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{r = cov/(s\_xs\_y)% Row Count 1 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{s\_x and s\_y are standard dev.}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Permutations}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{n! = n(n-1)(n-2)*...*1 if n≥1 \newline % Row Count 1 (+ 1)
~~~~= \seqsplit{1~~~~~~~~~~~~~~~~~~~~~~~~~if} n=0 \newline % Row Count 5 (+ 4)
nPr = n!/(n-r)!% Row Count 6 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{Order matters}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{PMF}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{f(x) = P(X=x)% Row Count 1 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Variance}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{σ\textasciicircum{}2\textasciicircum{} = V{[}X{]} = Σx\textasciicircum{}2\textasciicircum{}f(x)-E{[}X{]}\textasciicircum{}2\textasciicircum{}% Row Count 1 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{Standard deviation = sqrt(V{[}X{]})}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Joint Properties}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{E{[}g(X,Y){]} = Σ\textasciicircum{}x\textasciicircum{}Σ\textasciicircum{}y\textasciicircum{}g(x,y)p(x,y) \newline % Row Count 1 (+ 1)
E{[}X{]} = Σ\textasciicircum{}x\textasciicircum{}xp(x) \newline % Row Count 2 (+ 1)
E{[}Y{]} = Σ\textasciicircum{}y\textasciicircum{}yp(y) \newline % Row Count 3 (+ 1)
E{[}X+Y{]} = E{[}X{]}+E{[}Y{]} \newline % Row Count 4 (+ 1)
Cov{[}X,Y{]} = (Σ\textasciicircum{}x\textasciicircum{}Σ\textasciicircum{}y\textasciicircum{}xyp(x,y))-E{[}X{]}E{[}Y{]} \newline % Row Count 5 (+ 1)
V{[}X+Y{]} = V{[}X{]}+V{[}Y{]}+2Cov{[}X,Y{]}% Row Count 6 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Poisson Distribution}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{X = \# of event in time {[}0,1{]} \newline % Row Count 1 (+ 1)
p(x) = e\textasciicircum{}-μ\textasciicircum{}*μ\textasciicircum{}x\textasciicircum{}/x!, for x=0,1,... \newline % Row Count 2 (+ 1)
X \textasciitilde{} P(μ) \newline % Row Count 3 (+ 1)
E{[}X{]} = V{[}X{]} = μ \newline % Row Count 4 (+ 1)
Approximation: binomial f(x) ≈ p(x), μ=np \newline % Row Count 5 (+ 1)
Process: between {[}0,t{]}, μ=λt% Row Count 6 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Continuous Uniform Distribution}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{f(x) = 1/(b-a), a≤x≤b \newline % Row Count 1 (+ 1)
~~~~~~= 0, ~~~~~~~~~elsewhere \newline % Row Count 4 (+ 3)
X \textasciitilde{} U{[}a,b{]} \newline % Row Count 5 (+ 1)
E{[}X{]} = (a+b)/2 \newline % Row Count 6 (+ 1)
V{[}X{]} = (b-a)\textasciicircum{}2\textasciicircum{}/12% Row Count 7 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Sample Variance}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{s\textasciicircum{}2\textasciicircum{} = ((Σx\textasciicircum{}2\textasciicircum{}\_i)-nx̄\textasciicircum{}2\textasciicircum{})/(n-1)% Row Count 1 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{CLT}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{Z = (X̄-μ)/(σ/√n) \newline % Row Count 1 (+ 1)
X̄ \textasciitilde{} N(μ, σ\textasciicircum{}2\textasciicircum{}/n) ⇒ Z \textasciitilde{} N(0,1)% Row Count 2 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Confidence Level}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{α = P(Z\textgreater{}z\_α) = 1-Φ(z) \newline % Row Count 1 (+ 1)
μ ∈ {[}x̄-E, x̄+E{]} \newline % Row Count 2 (+ 1)
σ\textasciicircum{}2\textasciicircum{} known: E = z\_{[}α/2{]}*σ/√n \newline % Row Count 3 (+ 1)
σ\textasciicircum{}2\textasciicircum{} unknown: T = (X̄-μ)/(S/√n) \textasciitilde{} T(n-1) \newline % Row Count 4 (+ 1)
~~~~~~~~~~~~~~~~~~~~~P(T\textgreater{}t\_{[}α,v{]}) = α; z\_α = t\_{[}α,∞{]} \newline % Row Count 8 (+ 4)
~~~~~~~~~~~~~~~~~~~~~E = t\_{[}α/2,n-1{]}*s/√n \newline % Row Count 12 (+ 4)
σ\textasciicircum{}2\textasciicircum{} unknown, n≥40: (X̄-μ)/(S/√n) \textasciitilde{} N(0,1) \newline % Row Count 13 (+ 1)
\seqsplit{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~E} = z\_{[}α/2{]}*s/√n \newline % Row Count 18 (+ 5)
n≥((z\_{[}α/2{]}σ)/E)\textasciicircum{}2\textasciicircum{}% Row Count 19 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Combinations}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{n = n\_1*...*n\_k \newline % Row Count 1 (+ 1)
nCr = (\textasciicircum{}n\textasciicircum{}r) = n!/r!(n-r)!% Row Count 2 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{Order doesn't matter}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Multiplucation Rule}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{P(A∩B) = P(B|A)P(A) = P(A|B)P(B) \newline % Row Count 1 (+ 1)
~~~~~~~~~~~= P(A)P(B) if ind.% Row Count 3 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Transformation}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{E{[}g(X){]} = Σg(x)f(x) \newline % Row Count 1 (+ 1)
V{[}g(X){]} = {[}Σ(g(x))\textasciicircum{}2\textasciicircum{}f(x){]}-(E{[}g(X){]})\textasciicircum{}2\textasciicircum{}% Row Count 2 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Bernoulli Trial}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{S = \{success, failure\} = \{p,q\} \newline % Row Count 1 (+ 1)
p = P(I=1) \newline % Row Count 2 (+ 1)
I \textasciitilde{} Ber(p) \newline % Row Count 3 (+ 1)
E{[}I{]} = p \newline % Row Count 4 (+ 1)
V{[}I{]} = p(1-p)% Row Count 5 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Negative Binomial Distribution}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{X = \# of trials to until r\textasciicircum{}th\textasciicircum{} success \newline % Row Count 1 (+ 1)
X \textasciitilde{} Nb(r,p) \newline % Row Count 2 (+ 1)
f(x) = (\textasciicircum{}x-1\textasciicircum{}r-1)(1-p)\textasciicircum{}x-r\textasciicircum{}p\textasciicircum{}r\textasciicircum{}, for x=r,r+1,... \newline % Row Count 3 (+ 1)
E{[}X{]} = r/p \newline % Row Count 4 (+ 1)
V{[}X{]} = r(1-p)/p\textasciicircum{}2\textasciicircum{}% Row Count 5 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Erlang Distribution}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{T = time until r\textasciicircum{}th\textasciicircum{} outcome of Poisson process \newline % Row Count 1 (+ 1)
F(x) = P(T≤x) = 1-P(T\textgreater{}x) \newline % Row Count 2 (+ 1)
~~~~~~~= 1-Σ\textasciicircum{}r-1\textasciicircum{}e\textasciicircum{}-λx\textasciicircum{}(λx)\textasciicircum{}k\textasciicircum{}/k! \newline % Row Count 4 (+ 2)
E{[}T{]} = r/λ \newline % Row Count 5 (+ 1)
V{[}T{]} = r(1-λ)/λ\textasciicircum{}2\textasciicircum{}% Row Count 6 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Standardization Thm}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{Z = (X-E{[}X{]})/√(V{[}X{]}) \newline % Row Count 1 (+ 1)
F(x) = P(X≤x) = Ф((x-μ)/σ) \newline % Row Count 2 (+ 1)
P(a\textless{}X\textless{}b) = F(b)-F(a)% Row Count 3 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Percentile}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{Rank of k\textasciicircum{}th\textasciicircum{} percentile: (n+1)*k/100 = m+p, 0≤p\textless{}1 \newline % Row Count 2 (+ 2)
k\textasciicircum{}th\textasciicircum{} percentile = y\_m+p(y\_{[}m+1{]}-y\_m) \newline % Row Count 3 (+ 1)
IQR = q\_3-q\_1% Row Count 4 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{Median is 50\textasciicircum{}th\textasciicircum{} percentile}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Hypothesis}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{Null hyp: make no change \newline % Row Count 1 (+ 1)
Alternate hyp: test according to question \newline % Row Count 2 (+ 1)
⇒Test 1: μ ≠ μ\_0; 2: μ \textgreater{} μ\_0; 3: μ \textless{} μ\_0; \newline % Row Count 4 (+ 2)
Confidence interval decision: reject H\_0 for H\_1 if μ\_0 is not in confidence interval \newline % Row Count 6 (+ 2)
Z\_0 or T\_0 decision: \newline % Row Count 7 (+ 1)
σ\textasciicircum{}2\textasciicircum{} known: Z\_0 = (X̄-μ\_0)/(σ/√n) \textasciitilde{} N(0,1) \newline % Row Count 8 (+ 1)
Test 1: reject if |z\_0| \textgreater{} z\_{[}α/2{]}; 2: z\_0 \textgreater{} z\_α; 3: z\_0 \textless{} -z\_α \newline % Row Count 10 (+ 2)
σ\textasciicircum{}2\textasciicircum{} unknown: T\_0 = (X̄-μ\_0)/(S/√n) \textasciitilde{} T\_{[}n-1{]} \newline % Row Count 12 (+ 2)
Test 1: |t\_0| \textgreater{} t\_{[}α/2,n-1{]}; 2: t\_0 \textgreater{} t\_{[}α,n-1{]}; 3: t\_0 \textless{} -t\_{[}α,n-1{]} \newline % Row Count 14 (+ 2)
Pop. \& σ\textasciicircum{}2\textasciicircum{} unknown: replace σ with S from σ\textasciicircum{}2\textasciicircum{} known \newline % Row Count 16 (+ 2)
p-Value decision: reject if p-value \textless{} α \newline % Row Count 17 (+ 1)
p-value = 2{[}1-Ф(|z\_0|){]}, test 1 \& z-value \newline % Row Count 18 (+ 1)
= 1-Ф(z\_0), test 2 \& z-value \newline % Row Count 19 (+ 1)
= Ф(z\_0), test 3 \& z-value \newline % Row Count 20 (+ 1)
= 2P(T\textgreater{}|t\_0|), test 1 \& t-value \newline % Row Count 21 (+ 1)
= P(T\textgreater{}t\_0), test 2 \& t-value \newline % Row Count 22 (+ 1)
= P(T\textless{}t\_0), test 3 \& t-value% Row Count 23 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Addition Rules}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{P(A∩B') = P(A)-P(A∩B) \newline % Row Count 1 (+ 1)
P(A∪B) = P(A)+P(B)-P(A∩B) \newline % Row Count 2 (+ 1)
P(A'∩B') = 1-P(A∪B) \newline % Row Count 3 (+ 1)
P(A∪B∪C) = \seqsplit{P(A)+P(B)+P(C)-P(A∩B)-P(A∩C)-P(B∩C)+P(A∩B∩C)} \newline % Row Count 5 (+ 2)
P(A\_1∪...∪A\_n) = 1-P(A\_1'∩...∩A\_n')% Row Count 6 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Expected Value}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{μ = E{[}X{]} = Σxf(x)% Row Count 1 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Marginal PMF}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{p(x) = P(X=x) = Σ\textasciicircum{}y\textasciicircum{}p(x,y) \newline % Row Count 1 (+ 1)
p(y) = P(Y=y) = Σ\textasciicircum{}x\textasciicircum{}p(x,y)% Row Count 2 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Binomial Distribution}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{X = \# of successes from n trials \newline % Row Count 1 (+ 1)
X \textasciitilde{} b(n,p) \newline % Row Count 2 (+ 1)
f(x) = (\textasciicircum{}n\textasciicircum{}x)p\textasciicircum{}x\textasciicircum{}(1-p)\textasciicircum{}n-x\textasciicircum{}, for x=0,1,...,n \newline % Row Count 3 (+ 1)
E{[}X{]} = np \newline % Row Count 4 (+ 1)
V{[}X{]} = np(1-p)% Row Count 5 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Exponential Distribution}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{Waiting time \newline % Row Count 1 (+ 1)
X \textasciitilde{} Exp(λ) \newline % Row Count 2 (+ 1)
f(x) = λe\textasciicircum{}-λx\textasciicircum{}, x\textgreater{}0 \newline % Row Count 3 (+ 1)
F(x) = 1-e\textasciicircum{}-λx\textasciicircum{}, x\textgreater{}0 \newline % Row Count 4 (+ 1)
E{[}X{]} = 1/λ \newline % Row Count 5 (+ 1)
V{[}X{]} = 1/λ\textasciicircum{}2\textasciicircum{} \newline % Row Count 6 (+ 1)
Lack of memory: P(X\textgreater{}s+t|X\textgreater{}s) = P(X\textgreater{}t)% Row Count 7 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Standard Normal Distribution}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{Z \textasciitilde{} N(0,1) \newline % Row Count 1 (+ 1)
PMF: ⌀(z) = 1/√(2π)*e\textasciicircum{}-1/2*z\textasciicircum{}2\textasciicircum{} \newline % Row Count 2 (+ 1)
CDF: Φ(z) = P(Z≤z) = ∫⌀(t)dt \newline % Row Count 3 (+ 1)
Φ(0) = 0.5 \newline % Row Count 4 (+ 1)
P(Z≤-z) = P(Z≥z) \newline % Row Count 5 (+ 1)
Φ(-z) = 1-Φ(z) \newline % Row Count 6 (+ 1)
P(a≤Z≤b) = Φ(b)-Φ(a) \newline % Row Count 7 (+ 1)
P(-a≤Z≤-b) = Φ(a)-Φ(b)% Row Count 8 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Linear Combination}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{Y \textasciitilde{} N(μ\_Y, σ\textasciicircum{}2\textasciicircum{}\_Y) \newline % Row Count 1 (+ 1)
E{[}Y{]} = Σc\_iE{[}X\_i{]} \newline % Row Count 2 (+ 1)
V{[}Y{]} = Σc\textasciicircum{}2\textasciicircum{}\_iV{[}X\_i{]}\textasciicircum{}2\textasciicircum{} \newline % Row Count 3 (+ 1)
X̄ = 1/nΣX\_i \newline % Row Count 4 (+ 1)
E{[}X̄{]} = μ \newline % Row Count 5 (+ 1)
V{[}X̄{]} = σ\textasciicircum{}2\textasciicircum{}/n% Row Count 6 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{3.833cm}}{Y = c\_1X\_1+...+c\_nX\_n}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Sample Covariance}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{cov = \seqsplit{((Σx\_iy\_i)-(Σx\_i)(Σy\_i)/n)/(n-1)}% Row Count 1 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{3.833cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{3.833cm}}{\bf\textcolor{white}{Line of Best Fit}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{3.833cm}}{y = a+Bx \newline % Row Count 1 (+ 1)
B = ((Σx\_iy\_i)-(Σx\_i)(Σy\_i)/n)/((Σx\textasciicircum{}2\textasciicircum{}\_i)-(Σx\_i)\textasciicircum{}2\textasciicircum{}/n)% Row Count 3 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}