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% Document Info
\author{madsonic}
\pdfinfo{
  /Title (st2334.pdf)
  /Creator (Cheatography)
  /Author (madsonic)
  /Subject (ST2334 Cheat Sheet)
}

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\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{ST2334 Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{madsonic} via \textcolor{DarkBackground}{\uline{cheatography.com/21194/cs/3988/}}}
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\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}madsonic \\
  \uline{cheatography.com/madsonic} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 27th April, 2015.\\
   Updated 13th May, 2016.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




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\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{17.67cm}}{\bf\textcolor{white}{Definitions}}  \tn
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Sample Space & The set of all possible outcomes of an \seqsplit{experiment} is called the sample space and is denoted by Ω. &  &  \tn 
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Sigma field & A \seqsplit{collection} of sets F of Ω is called a σ-field if it satisfies the following \seqsplit{conditions:} &  &  \tn 
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 & 1. ∅ ∈ F ~~~~2. If A1,...,∈ F then 􏰍U∞1 Ai ∈ F &  & 3. If A ∈ F then A\textasciicircum{}c\textasciicircum{} ∈ F \tn 
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\seqsplit{Probability} & A \seqsplit{probability} measure P on (Ω, F ) is a function P : F → {[}0, 1{]} which \seqsplit{satisfies:} &  &  \tn 
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 & 1.P(Ω)=1 and P(∅)=0 & 2. &  \tn 
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\seqsplit{Conditional} \seqsplit{Probability} & Consider \seqsplit{probability} space (Ω, F , P) and let A, B ∈ F with P(B) \textgreater{} 0. Then the \seqsplit{conditional} \seqsplit{probability} that A occurs given B occurs is defined to be: P(A|B) = P(A ∩ B) / P(B) &  &  \tn 
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Total \seqsplit{Probability} & A family of sets B1, . ., Bn is called a partition of Ω if: ∀i !=j Bi ∩Bj =∅ and 􏰒U∞1 Bi =Ω & P(A) = ∑n1 \seqsplit{P(A|Bi)P(Bi)} & P(A) = ∑n1 P(A∩Bi) \tn 
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\seqsplit{Independence} & Consider \seqsplit{probability} space (Ω, F , P) and let A, B ∈ F . A and B are \seqsplit{independent} if P(A ∩ B) = P(A)P(B) &  &  \tn 
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 & More generally, a family of F−sets A1,...,An (∞ \textgreater{} n ≥ 2) are \seqsplit{independent} \seqsplit{if􏰃􏰓􏰄􏰐} P(∩n1 Ai) = ∏ n1 P(Ai) &  &  \tn 
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Random Variable (RV) & A RV is a function X : Ω → R such that for each x ∈ R, \{ω ∈ Ω : X(ω) ≤ x\} ∈ F. Such a function is said to be \seqsplit{F−measurable} &  &  \tn 
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\seqsplit{Distribution} Function & \seqsplit{Distribution} function of a random variable X is the function F : R → {[}0, 1{]} given by F(x)=P(X ≤x), x∈R. &  &  \tn 
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Discrete RV & A RV is said to be discrete if it takes values in some countable subset X = \{x1,x2,...\} of R &  &  \tn 
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PMF & PMF of a discrete RV X, is the function f :X→{[}0,1{]} defined by f(x)=P(X =x). It satisfy: & PDF & function f is called the \seqsplit{probability} density function (PDF) of the con- tinuous random variable X \tn 
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 & 1. set of x s.t. f(x) != 0 is countable &  & f(x) = F'(x) \tn 
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 & 2. \seqsplit{∑􏰊x∈X} f(x) = 1 &  & F(x) = ∫-∞x f(u) du \tn 
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 & 3. f(x) ≥ 0 &  &  \tn 
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\seqsplit{Independence} & Discrete RV X and Y are indie if the events \{X = x\} \& \{Y =y\} are indie for each(x,y)∈X×Y & The RV X and Y are indie if \{X≤x\} \{Y≤y\} are indie events for each x, y ∈ R &  \tn 
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 & P(X,Y) = \seqsplit{P(X=x)P(Y=y)} &  &  \tn 
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 & f(x,y) = f(x)f(y) & f(x,y) = f(x)f(y) F(x,y)  v &  \tn 
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 & E{[}XY{]} = E{[}X{]}E{[}Y{]} &  &  \tn 
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\seqsplit{Expectation} & expected value of RV {\emph{X}} on X, & The \seqsplit{expectation} of a \seqsplit{continuous} random variable X with PDF f is given by &  \tn 
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 & E{[}{\emph{X}}{]} = ∑x∈X xf(x) & E{[}{\emph{X}}{]} = ∫x∈X xf(x) dx &  \tn 
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 & E{[}g(x){]} = ∑x∈X g(x)f(x) & E{[}g(x){]} = ∫x∈X g(x)f(x) dx &  \tn 
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Variance & spread of RV & E{[}({\emph{X}} − E{[}{\emph{X}}{]})\textasciicircum{}2\textasciicircum{}{]} & E{[}{\emph{X}}\textasciicircum{}2\textasciicircum{}{]} - E{[}{\emph{X}}{]}\textasciicircum{}2\textasciicircum{} \tn 
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MGF (uniquely \seqsplit{characterises} \seqsplit{distribution)} & M(t) = E{[}e\textasciicircum{}{\emph{X}}t\textasciicircum{}{]} = ∑x∈X e\textasciicircum{}{\emph{X}}t\textasciicircum{} f(x) & t∈T s.t. t for ∑x∈X e\textasciicircum{}{\emph{X}}t\textasciicircum{} f(x) \textless{} ∞ &  \tn 
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 & M(t) = E{[}e\textasciicircum{}{\emph{X}}t\textasciicircum{}{]} = ∫x∈X e\textasciicircum{}{\emph{X}}t\textasciicircum{} f(x) dx & t∈T s.t. t for ∫x∈X e\textasciicircum{}{\emph{X}}t\textasciicircum{} f(x) dx \textless{} ∞ &  \tn 
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 & M(t1,t2) = E{[}e\textasciicircum{}Xt1+Yt2\textasciicircum{}{]} = ∫z  e\textasciicircum{}Xt1+Yt2\textasciicircum{} f(x,y) dxdy (t1,t2)∈T & E{[}X{]} = ∂/∂t1 M(t1,t2) |t1=t2=0 & E{[}XY{]} = ∂\textasciicircum{}2\textasciicircum{}/∂t1∂t2 M(t1,t2) |t1=t2=0 \tn 
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 & E{[}{\emph{X\textasciicircum{}k\textasciicircum{}}}{]} = M\textasciicircum{}k\textasciicircum{}(0) &  &  \tn 
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Moment & Given a discrete RV {\emph{X}} on X, with PMF f and k ∈ Z\textasciicircum{}+\textasciicircum{}, the k\textasciicircum{}th\textasciicircum{} moment of X is & E{[}X\textasciicircum{}k\textasciicircum{}{]} &  \tn 
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Central Moment & k\textasciicircum{}th\textasciicircum{} central moment of X is & E{[}({\emph{X}} − E{[}{\emph{X}}{]})\textasciicircum{}k\textasciicircum{}{]} &  \tn 
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\seqsplit{Dependence} & Joint \seqsplit{distribution} function F : R\textasciicircum{}2\textasciicircum{} → {[}0,1{]} of X,Y where X and Y are discrete random variables, is given by F(x,y) = \seqsplit{P(X≤x∩Y≤y)} & The joint \seqsplit{distribution} function of X and Y is the function F : R2 → {[}0, 1{]} given by F(x,y)=P(X≤x,Y ≤y) &  \tn 
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 & Joint mass function f : R2 → {[}0, 1{]} is given by f(x,y) = P(x∩y) & The random variables are jointly \seqsplit{continuous} with joint PDF f : R2 → {[}0, ∞) if F(x, y) = \seqsplit{∫-∞y∫-∞x} f(u,v) dudv &  \tn 
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 &  & f(x,y) = ∂\textasciicircum{}2\textasciicircum{}/∂x∂y F(x,y) &  \tn 
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Marginal & f(x) = ∑y∈Y f(x,y) & f(x) = ∫y∈Y f(x,y)dy & F(x) = lim y-\textgreater{}∞ F(x,y) F(x) = \seqsplit{∫-∞x∫-∞∞} f(u,y) dydu \tn 
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 & E{[}{\emph{g(x,y)}}{]} = ∑x,y∈{\emph{X}}x{\emph{Y}} g(x,y)f(x,y) & E{[}{\emph{g(x,y)}}{]} = ∫x,y∈{\emph{X}}x{\emph{Y}} g(x,y)f(x,y) dxdy &  \tn 
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\seqsplit{Covariance} & indie =\textgreater{} E{[}XY{]} = E{[}X{]}E{[}Y{]}, Cov = 0 =\textgreater{} ρ = 0 & ρ = 0 =\textgreater{} E{[}XY{]} = E{[}X{]}E{[}Y{]} &  \tn 
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 & Cov{[}X,Y{]} = E{[}(X − E{[}X{]})(Y − E{[}Y{]}){]} & Cov{[}X,Y{]} = E{[}XY{]} - E{[}X{]}E{[}Y{]} &  \tn 
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\seqsplit{Correlation} & Gives linear \seqsplit{relationship} (+/-). |ρ| close to 1 is strong, close to 0 is weak & special for \seqsplit{bi-variate} normal, indie \textless{}=\textgreater{} \seqsplit{uncorrelated} &  \tn 
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 & ρ(X,Y)= Cov{[}X,Y{]} / sqrt(􏰗Var{[}X{]}Var{[}Y{]}) &  &  \tn 
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\seqsplit{Conditional} \seqsplit{distribution} & The \seqsplit{conditional} \seqsplit{distribution} function of Y given X, written FY |x(·|x), is defined by & F(y|x) = ∫-∞y f(x,v)/f(x) dv & f(y|x) = f(x,y)/f(x) where f(x) = \seqsplit{∫-∞∞} f(x,y) dy \tn 
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 & Fy|x(y|x) = P(Y ≤ y|X = x) &  &  \tn 
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 & for any x with P(X =x)\textgreater{}0. The \seqsplit{conditional} PMF of Y given X =x is defined by ... when x is s.t. P(X =x)\textgreater{}0 &  &  \tn 
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 & f(y|x) = P(Y = y|X = x) &  &  \tn 
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 & f(x,y) = \seqsplit{f(x|y)f(y)} or \seqsplit{f(y|x)f(x)} &  &  \tn 
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\seqsplit{Conditional} \seqsplit{expectation} & The \seqsplit{conditional} \seqsplit{expectation} of a RV Y, given X = x is E{[}Y|X =x{]} = \seqsplit{􏰏∑y∈Y} yf(y|x) given that the \seqsplit{conditional} PMF is \seqsplit{well-defined} & E{[}h(X)g(Y){]} = E{[}E{[}g(Y)|X{]}h(X){]} = \seqsplit{∫(∫g(Y)f(Y|X)} dx) h(X)f(x) dx &  \tn 
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 & E{[}Y|X =x{]} = \seqsplit{􏰏∑y∈Y} yf(y|x) & E{[}E{[}Y|X{]}{]} = E{[}Y{]} & E{[}E{[}Y|X{]}g(X){]} = E{[}Yg(X){]} \tn 
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 & E{[}(aX + bY)|Z{]} = aE{[}X|Z{]} + bE{[}Y|Z{]} &  &  \tn 
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 & if X and Y are \seqsplit{independent} & E{[}X|Y{]} = E{[}X{]} & Var{[}X|Y{]} = E{[}X\textasciicircum{}2\textasciicircum{}|Y{]} - E{[}X|Y{]}\textasciicircum{}2\textasciicircum{} \tn 
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Bayes Theorem & Consider probability space (Ω, F , P) and let A, B ∈ F with P(A), P(B) \textgreater{} 0. Then we have: \tn 
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 & P(B|A) = P(A|B)P(B) / P(A) \tn 
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Independence & If X and Y are indie RV and g : X → R, h : Y → R, then the RV g(X) and h(Y ) are also indie \tn 
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Expectations & 1. if X≥0, E{[}X{]}≥0 \tn 
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 & 2. if a, b∈R then E{[}aX+bY{]}=aE{[}X{]}+bE{[}Y{]} \tn 
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 & 3. if X = c∈R always, then E{[}X{]}=c. \tn 
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Variance & 1. For a ∈ R, Var{[}aX{]} = a\textasciicircum{}2\textasciicircum{}Var{[}X{]} \tn 
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 & 2. Uncorrelated Var{[}X + Y{]} = Var{[}X{]} + Var{[}Y{]} \tn 
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Conditional Expectation & Conditional expectations satisfies E{[}E{[}Y|X{]}{]} = E{[}Y{]} assuming all the expectations exist \tn 
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 & for any g : R → R, E{[}E{[}Y|X{]}g(X){]} = E{[}Yg(X){]} assuming all expectations exist \tn 
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Change of variable & If (X1,X2) have joint density f(x,y) on Z, then for (Y1,Y2) = T(X1,X2), with T as described above, the joint density of (Y1,Y2), denoted g is: g(y1,y2)=f(T\textasciicircum{}−1\textasciicircum{}(y1,y2),T\textasciicircum{}−1\textasciicircum{}(y1,y2)) |J(y ,y )| (y1,y2)∈T \tn 
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\end{tabularx}
\par\addvspace{1.3em}



\end{document}