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\author{madsysharma}
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  /Title (probability-midterm.pdf)
  /Creator (Cheatography)
  /Author (madsysharma)
  /Subject (Probability - Midterm Cheat Sheet)
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\noindent
\begin{multicols}{3}
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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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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Probability - Midterm Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{madsysharma} via \textcolor{DarkBackground}{\uline{cheatography.com/208834/cs/44841/}}}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}madsysharma \\
  \uline{cheatography.com/madsysharma} \\
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 11th December, 2024.\\
   Page {\thepage} of \pageref{LastPage}.
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  Measure your website readability!\\
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\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{x{1.00248 cm} p{0.66832 cm} x{1.2531 cm} x{1.2531 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{5.377cm}}{\bf\textcolor{white}{Special Distributions (Discrete RVs)}}  \tn
% Row 0
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{\bf{E and Var}} & {\bf{NAME}} & {\bf{R\textasciitilde{}X\textasciitilde{}}} & {\bf{PMF}} \tn 
% Row Count 2 (+ 2)
% Row 1
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p \& p(1-p) & \seqsplit{Bernoulli(p)} & \{0,1\} & p for x=1, 1-p for x=0 \tn 
% Row Count 4 (+ 2)
% Row 2
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1/p and (1-p)/p\textasciicircum{}2\textasciicircum{} & \seqsplit{Geometric(p)} & Z\textasciicircum{}+\textasciicircum{} & p(1-p)\textasciicircum{}(k-1)\textasciicircum{} for k \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} Z\textasciicircum{}+\textasciicircum{} \tn 
% Row Count 7 (+ 3)
% Row 3
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np and np(1-p) & \seqsplit{Binomial(n},p) & \{0,1,..,n\} & \textasciicircum{}n\textasciicircum{}C\textasciitilde{}k\textasciitilde{} . p\textasciicircum{}k\textasciicircum{} . (1-p)\textasciicircum{}(n-k)\textasciicircum{}  for k = 0 to n \tn 
% Row Count 11 (+ 4)
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m/p and (m.(1-p))/p\textasciicircum{}2\textasciicircum{} & \seqsplit{Pascal(m},p) & \{m,m+1,m+2...\} & \textasciicircum{}(k-1)\textasciicircum{}C\textasciitilde{}(m-1)\textasciitilde{} . p\textasciicircum{}m\textasciicircum{} . (1-p)\textasciicircum{}(k-m)\textasciicircum{} for k = m, m+1, m+2, m+3, ... \tn 
% Row Count 17 (+ 6)
% Row 5
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np and \seqsplit{((b+r-n)/(b+r-1))}.np(1-p) & \seqsplit{Hypergeometric(b},r,k) & \{max(0,k-r), max(0,k-r)+1, ..., min(k,b)\} & (\textasciicircum{}b\textasciicircum{}C\textasciitilde{}x\textasciitilde{} . \textasciicircum{}r\textasciicircum{}C\textasciitilde{}(k-x)\textasciitilde{})/(\textasciicircum{}(b+r)\textasciicircum{}C\textasciitilde{}k\textasciitilde{}) \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R\textasciitilde{}X\textasciitilde{} \tn 
% Row Count 22 (+ 5)
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Both equal to lambda & \seqsplit{Poisson(lambda)} & Z\textasciicircum{}+\textasciicircum{} & (e\textasciicircum{}-lambda\textasciicircum{} . lambda\textasciicircum{}k\textasciicircum{})/k! for k \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R\textasciitilde{}X\textasciitilde{} \tn 
% Row Count 26 (+ 4)
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Continuous RVs, PDFs and Mixed RVs}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{RV X with CDF F\textasciitilde{}X\textasciitilde{}(x) is continuous if F\textasciitilde{}X\textasciitilde{}(x) is a continuous function \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R \newline % Row Count 2 (+ 2)
PMF doesn't work for CRVs, since \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R, P\textasciitilde{}X\textasciitilde{}(x) = 0. Instead, PDFs are used. \newline % Row Count 4 (+ 2)
PDF = f\textasciitilde{}X\textasciitilde{}(x) = dF\textasciitilde{}X\textasciitilde{}(x)/dx (if F\textasciitilde{}X\textasciitilde{}(x) is differentiable at x) \textgreater{}=0  \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R.  \newline % Row Count 6 (+ 2)
{\bf{P(a\textless{}X\textless{}=b) = integral from a to b (f\textasciitilde{}X\textasciitilde{}(u).du)}} and {\bf{integral from -inf to +inf (f\textasciitilde{}X\textasciitilde{}(u) . du) = 1}} \newline % Row Count 9 (+ 3)
{\bf{EX = integral from -inf to +inf (x . f\textasciitilde{}X\textasciitilde{}(x) . dx)}} and {\bf{E{[}g(X){]} = integral from -inf to +inf (g(x) . f\textasciitilde{}X\textasciitilde{}(x) . dx)}} \newline % Row Count 12 (+ 3)
{\bf{Var(X) = integral from -inf to +inf (x\textasciicircum{}2\textasciicircum{} . f\textasciitilde{}X\textasciitilde{}(x) . dx - mu\textasciicircum{}2\textasciicircum{}\textasciitilde{}X\textasciitilde{})}} \newline % Row Count 14 (+ 2)
If g: R-\textgreater{} R is strictly monotonic and differentiable, then {\bf{PDF of Y=g(X) is f\textasciitilde{}Y\textasciitilde{}(y) = f\textasciitilde{}X\textasciitilde{}(x\textasciitilde{}1\textasciitilde{}) . |dx\textasciitilde{}1\textasciitilde{}/dy| where g(x\textasciitilde{}1\textasciitilde{})=y and 0 if g(x) = y has no solution}}% Row Count 18 (+ 4)
} \tn 
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Joint Distributions: RVs \textgreater{}= 2}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Joint PMF of X and Y = P\textasciitilde{}XY\textasciitilde{}(x,y) = P(X=x, Y=y) = P((X=x) and (Y=y))}} and {\bf{Joint range = R\textasciitilde{}XY\textasciitilde{} = \{(x,y)| P\textasciitilde{}XY\textasciitilde{}(x,y) \textgreater{} 0\}}} and {\bf{summing up P\textasciitilde{}XY\textasciitilde{} over all (x,y) pairs will result in 1}} \newline % Row Count 4 (+ 4)
{\bf{Marginal PMF of X = P\textasciitilde{}X\textasciitilde{}(x) = sum over all y\textasciitilde{}j\textasciitilde{} \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R\textasciitilde{}Y\textasciitilde{} (P\textasciitilde{}XY\textasciitilde{}(x, y\textasciitilde{}j\textasciitilde{})) for any x \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R\textasciitilde{}X\textasciitilde{}}}. Similarly, {\bf{Marginal PMF of Y = P\textasciitilde{}Y\textasciitilde{}(y) = sum over all x\textasciitilde{}i\textasciitilde{} \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R\textasciitilde{}X\textasciitilde{} (P\textasciitilde{}XY\textasciitilde{}(x\textasciitilde{}i\textasciitilde{}, y)) for any y \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R\textasciitilde{}Y\textasciitilde{}}} \newline % Row Count 9 (+ 5)
To show independence between X and Y, {\bf{prove P(X = x, Y=y) = P(X=x) . P(Y=y) for all x-y pairs}}. Similarly, {\bf{for conditional independence, show that P(Y=y|X=x) = P(Y=y) for all x-y pairs}} \newline % Row Count 13 (+ 4)
{\bf{Joint CDF = F\textasciitilde{}XY\textasciitilde{}(x,y) = P(X\textless{}=x, Y\textless{}=y)}} and {\bf{Marginal CDF for X = F\textasciitilde{}X\textasciitilde{}(x) = limit y to inf(F\textasciitilde{}XY\textasciitilde{}(x,y)) for any x}} and {\bf{Marginal CDF for Y = F\textasciitilde{}Y\textasciitilde{}(y) = limit x to inf (F\textasciitilde{}XY\textasciitilde{}(x,y)) for any y}} \newline % Row Count 17 (+ 4)
{\bf{Conditional expectation: E{[}X|Y=y\textasciitilde{}j\textasciitilde{}{]} = sum over all x\textasciitilde{}i\textasciitilde{} \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R\textasciitilde{}X\textasciitilde{} (x\textasciitilde{}i\textasciitilde{} . P\textasciitilde{}X|Y\textasciitilde{}(x\textasciitilde{}i\textasciitilde{}|y\textasciitilde{}j\textasciitilde{}))}} \newline % Row Count 19 (+ 2)
{\bf{NOTE: F\textasciitilde{}XY\textasciitilde{}(inf,inf) = 1, F\textasciitilde{}XY\textasciitilde{}(-inf,y) = 0 for any y and F\textasciitilde{}XY\textasciitilde{}(x,-inf) = 0 for any x}} \newline % Row Count 21 (+ 2)
{\bf{P(x\textasciitilde{}1\textasciitilde{} \textless{} X \textless{}= x\textasciitilde{}2\textasciitilde{}, y\textasciitilde{}1\textasciitilde{} \textless{} Y \textless{}= y\textasciitilde{}2\textasciitilde{}) = F\textasciitilde{}XY\textasciitilde{}(x\textasciitilde{}1\textasciitilde{},y\textasciitilde{}1\textasciitilde{}) + F\textasciitilde{}XY\textasciitilde{}(x\textasciitilde{}2\textasciitilde{},y\textasciitilde{}2\textasciitilde{}) - F\textasciitilde{}XY\textasciitilde{}(x\textasciitilde{}2\textasciitilde{},y\textasciitilde{}1\textasciitilde{}) - F\textasciitilde{}XY\textasciitilde{}(x\textasciitilde{}1\textasciitilde{},y\textasciitilde{}2\textasciitilde{})}} \newline % Row Count 24 (+ 3)
{\bf{Conditional PMF given event A = P\textasciitilde{}X|A\textasciitilde{}(x\textasciitilde{}i\textasciitilde{}) = P(X=x\textasciitilde{}i\textasciitilde{}|A) = P(X=x\textasciitilde{}i\textasciitilde{} and A)/P(A) for any x\textasciitilde{}i\textasciitilde{} \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R\textasciitilde{}X\textasciitilde{}}} and {\bf{Conditional CDF = F\textasciitilde{}X|A\textasciitilde{}(x) = P(X \textless{}= x | A)}} \newline % Row Count 28 (+ 4)
{\bf{Given RVs X and Y, P\textasciitilde{}X|Y\textasciitilde{}(x\textasciitilde{}i\textasciitilde{}, y\textasciitilde{}j\textasciitilde{}) = P\textasciitilde{}XY\textasciitilde{}(x\textasciitilde{}i\textasciitilde{},y\textasciitilde{}j\textasciitilde{})/P\textasciitilde{}Y\textasciitilde{}(y\textasciitilde{}j\textasciitilde{}). Similarly for Y|X}} \newline % Row Count 30 (+ 2)
} \tn 
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Joint Distributions: RVs \textgreater{}= 2 (cont)}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{{\bf{E{[}X + Y{]} = E{[}X{]} + E{[}Y{]} - independence not required}} \newline % Row Count 2 (+ 2)
{\bf{E{[}X . Y{]} = E{[}X{]} . E{[}Y{]} - independence IS required}}% Row Count 4 (+ 2)
} \tn 
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Problem Solving Techniques}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{{\bf{* CARD PROBLEMS: }} \newline % Row Count 1 (+ 1)
Number of ways to pick k suits = \textasciicircum{}4\textasciicircum{}C\textasciitilde{}k\textasciitilde{} with k=1,2,3,4 \newline % Row Count 3 (+ 2)
{\bf{* n BALLS, r BINS: }}  \newline % Row Count 4 (+ 1)
- Distinguishable balls: each ball can go into any 1 of r bins. The \# of distinct perms would be \textasciicircum{}r\textasciicircum{}\textasciitilde{}\textasciitilde{}P\textasciitilde{}\textasciitilde{}\textasciitilde{}n\textasciitilde{} = r\textasciicircum{}n\textasciicircum{} \newline % Row Count 7 (+ 3)
- Indistinguishable balls: there will be 2 cases: \newline % Row Count 8 (+ 1)
  * No empty bins. Occupancy vector is x\textasciitilde{}1\textasciitilde{}+...+x\textasciitilde{}r\textasciitilde{}=n where every x is \textgreater{}= 1. There can be n-1 possible locations for bin dividers from which we can choose r-1 to keep \textgreater{}= 1 ball in each bin. \# of possible arrangements = \textasciicircum{}(n-1)\textasciicircum{}C\textasciitilde{}(r-1)\textasciitilde{}. \newline % Row Count 13 (+ 5)
  * Bin may have 0 balls. Then the occupancy vector would be y\textasciitilde{}1\textasciitilde{}+...+y\textasciitilde{}r\textasciitilde{}=n+r and the \# of arrangements will be \textasciicircum{}(n+r-1)\textasciicircum{}C\textasciitilde{}(r-1)\textasciitilde{} \newline % Row Count 16 (+ 3)
{\bf{* COMMITTEE SELECTION: Solve using product rule/hypergeometric approach.}} \newline % Row Count 18 (+ 2)
{\bf{* HAT MATCHING PROBLEM:}} \newline % Row Count 19 (+ 1)
\{\{fa-chevron-right\}\} Probability of k men drawing their own hats (over all k-tuples) = {\bf{(\textasciicircum{}n\textasciicircum{}C\textasciitilde{}k\textasciitilde{}(n-k)!)/n! = 1/k!}} \newline % Row Count 22 (+ 3)
\# of derangements = {\bf{n!{[}1-1/1!+1/2!-1/3!+...+(-1)\textasciicircum{}n\textasciicircum{}/n!{]}}} \newline % Row Count 24 (+ 2)
\{\{fa-chevron-right\}\} P(k matches) = {\bf{{[}1/2! - 1/3! + 1/4! -...+(-1)\textasciicircum{}(n-k)\textasciicircum{}/(n-k)!{]}/k!}} \newline % Row Count 26 (+ 2)
{\bf{* DRAWING THE ONLY SPECIAL BALL FROM n BALLS IN k TRIALS:}} \newline % Row Count 28 (+ 2)
Total \# of outcomes = {\bf{\textasciicircum{}n\textasciicircum{}C\textasciitilde{}k\textasciitilde{} = \textasciicircum{}{[}1 + (n-1){]}\textasciicircum{}C\textasciitilde{}k\textasciitilde{} = \textasciicircum{}1\textasciicircum{}C\textasciitilde{}0\textasciitilde{}\textasciicircum{}(n-1)\textasciicircum{}C\textasciitilde{}k\textasciitilde{} + \textasciicircum{}1\textasciicircum{}C\textasciitilde{}1\textasciitilde{}\textasciicircum{}(n-1)\textasciicircum{}C\textasciitilde{}(k-1)\textasciitilde{} , with term \#1 denoting no special ball, and term \#2 denoting the special ball}} \newline % Row Count 32 (+ 4)
} \tn 
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Problem Solving Techniques (cont)}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{{\bf{*Total \# of roundtable arrangements with k people = k!/k = (k-1)!}} \newline % Row Count 2 (+ 2)
{\bf{* SYSTEM RELIABILITY ANALYSIS:}} \newline % Row Count 3 (+ 1)
\{\{fa-chevron-right\}\} P(fail)=p, P(success)=1-p \newline % Row Count 4 (+ 1)
\{\{fa-chevron-right\}\} For parallel config, 2\textasciicircum{}n\textasciicircum{}-1 successes and 1 failure, P(fail)=p\textasciicircum{}n\textasciicircum{} \newline % Row Count 6 (+ 2)
\{\{fa-chevron-right\}\} For series config, 2\textasciicircum{}n\textasciicircum{}-1 failures and 1 success, P(success) = (1-p)\textasciicircum{}n\textasciicircum{} \newline % Row Count 8 (+ 2)
\{\{fa-chevron-right\}\} For series connections, take intersection, and for parallel connections take union \newline % Row Count 11 (+ 3)
{\bf{* PMF FOR SUM, DIFF, MAX, MIN OF 4-SIDED DICE:}} \newline % Row Count 13 (+ 2)
\{\{fa-chevron-right\}\} Uniform PMF = P\textasciitilde{}XY\textasciitilde{}(x,y) = 1/16 \newline % Row Count 15 (+ 2)
\{\{fa-chevron-right\}\} For each (x,y) point in the Cartesian coordinate diagram, calculate the diff/sum label or min/max label. \newline % Row Count 18 (+ 3)
\{\{fa-chevron-right\}\} Write down tables for Joint, Marginal and Conditional PMFs \newline % Row Count 20 (+ 2)
\{\{fa-chevron-right\}\} Headers are: x y P\textasciitilde{}XY\textasciitilde{}(x,y) x P\textasciitilde{}X\textasciitilde{}(x) y P\textasciitilde{}Y\textasciitilde{}(y) x y P\textasciitilde{}Y|X\textasciitilde{}(y,x). First 3 for joint, next 4 for marginal, the remaining for conditional \newline % Row Count 24 (+ 4)
\{\{fa-chevron-right\}\} For marginal, plot PMF on y-axis and RV value on x-axis. \newline % Row Count 26 (+ 2)
\{\{fa-chevron-right\}\} For joint, plot y on y-axis and x on x-axis% Row Count 28 (+ 2)
} \tn 
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Facts for PMFs and RV Distributions}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{0≤P\textasciitilde{}X\textasciitilde{}(x)≤1 \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x}} and {\bf{Sum over all x \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R\textasciitilde{}x\textasciitilde{} (P\textasciitilde{}X\textasciitilde{}(x)) = 1}} \newline % Row Count 2 (+ 2)
{\bf{For any set A⊂R\textasciitilde{}X\textasciitilde{}, P(X∈A) = ∑\textasciitilde{}x∈A\textasciitilde{}P\textasciitilde{}X\textasciitilde{}(x)}} \newline % Row Count 4 (+ 2)
RVs X and Y are {\bf{independent if P(X=x, Y=y) = P(X=x) * P(Y=y), \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x,y}} The first formula can be extended to n times. \newline % Row Count 7 (+ 3)
{\bf{P(Y=y|X=x) = P(Y=y), \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x,y if X \& Y are independent}} \newline % Row Count 9 (+ 2)
If X\textasciitilde{}1\textasciitilde{},...,X\textasciitilde{}n\textasciitilde{} are independent Bernoulli(p) RVs, then X=X\textasciitilde{}1\textasciitilde{}+X\textasciitilde{}2\textasciitilde{}+...+X\textasciitilde{}n\textasciitilde{} has Binomial(n,p) distribution, and {\bf{Pascal (1,p) = Geometric (p)}} \newline % Row Count 12 (+ 3)
For distributions using parameter p, {\bf{0\textless{}p\textless{}1}} \newline % Row Count 13 (+ 1)
If X is of Binomial (n, p = lambda/n), with fixed lambda \textgreater{} 0. Then, {\bf{for any k \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} Z, lim\textasciitilde{}n-\textgreater{}inf\textasciitilde{}P\textasciitilde{}X\textasciitilde{}(k) = (e\textasciicircum{}-lambda\textasciicircum{}.lambda\textasciicircum{}k\textasciicircum{})/k!}} \newline % Row Count 16 (+ 3)
{\bf{* SPECIAL DISTRIBUTIONS:}} \newline % Row Count 17 (+ 1)
TYPE, PDF \& E{[}X{]} AND VAR(X) \newline % Row Count 18 (+ 1)
Uniform(a, b) {\bf{||}} 1/(b-a) if a\textless{}x\textless{}b {\bf{||}} (a+b)/2 and (b-a)\textasciicircum{}2\textasciicircum{}/12 \newline % Row Count 20 (+ 2)
Exponential(lambda) {\bf{||}} lambda . e\textasciicircum{}(-lambda . x)\textasciicircum{} {\bf{||}} 1/lambda and 1/(lambda)\textasciicircum{}2\textasciicircum{} \newline % Row Count 22 (+ 2)
Normal/Gaussian, ie: N(0,1) {\bf{||}} (1/sqrt(2 . pi)) . exp(-x\textasciicircum{}2\textasciicircum{}/2), \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R {\bf{||}} 0 and 1 \newline % Row Count 24 (+ 2)
Gamma (alpha, lambda) {\bf{||}} (lambda\textasciicircum{}alpha\textasciicircum{} . x\textasciicircum{}(alpha -1)\textasciicircum{} . e\textasciicircum{}(-lambda . x)\textasciicircum{})/(alpha -1)! for x\textgreater{}0 {\bf{||}} alpha/lambda and EX/lambda \newline % Row Count 27 (+ 3)
{\bf{CDF: F\textasciitilde{}X\textasciitilde{}(x) = P(X \textless{}= x) \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R}} and {\bf{P(a \textless{} X \textless{}= b) = F\textasciitilde{}X\textasciitilde{}(b) - F\textasciitilde{}X\textasciitilde{}(a)}}% Row Count 29 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Counting Principles, n-nomial Expansions}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Permutations of n distinct objs. take n w/ r groups of indistinct objs. = {\bf{(n!)/(n\textasciitilde{}1\textasciitilde{}! {\emph{ ... }} n\textasciitilde{}r\textasciitilde{}!)}} \newline % Row Count 3 (+ 3)
{\bf{\textasciicircum{}n\textasciicircum{}\textasciitilde{}\textasciitilde{}P\textasciitilde{}\textasciitilde{}\textasciitilde{}r\textasciitilde{} = n\textasciicircum{}r\textasciicircum{} and \textasciicircum{}n\textasciicircum{}\textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{}\textasciitilde{}r\textasciitilde{}=\textasciicircum{}(n+r-1)\textasciicircum{}C\textasciitilde{}r\textasciitilde{} : for perms and combs where k objs are taken at a time}} \newline % Row Count 6 (+ 3)
{\bf{(a+b)\textasciicircum{}n\textasciicircum{} = Sum over k (\textasciicircum{}n\textasciicircum{}C\textasciitilde{}k\textasciitilde{}a\textasciicircum{}k\textasciicircum{}b\textasciicircum{}(n-k)\textasciicircum{}) where k=0,...,n}} \newline % Row Count 8 (+ 2)
{\bf{Binomial coeff. identity: \textasciicircum{}n\textasciicircum{}C\textasciitilde{}k\textasciitilde{} = \textasciicircum{}(n-1)\textasciicircum{}C\textasciitilde{}(k-1)\textasciitilde{} +  \textasciicircum{}(n-1)\textasciicircum{}C\textasciitilde{}k\textasciitilde{} where first term maps to A and second to A\textasciicircum{}C\textasciicircum{}}} \newline % Row Count 11 (+ 3)
{\bf{ \textasciicircum{}n\textasciicircum{}C\textasciitilde{}m\textasciitilde{} = \textasciicircum{}n\textasciicircum{}C\textasciitilde{}n-m\textasciitilde{}}} \newline % Row Count 12 (+ 1)
{\bf{Sum over r (\textasciicircum{}n\textasciicircum{}C\textasciitilde{}r\textasciitilde{}(-1)\textasciicircum{}r\textasciicircum{}(1)\textasciicircum{}(n+r)\textasciicircum{} is 0 where r=0,...n}} \newline % Row Count 14 (+ 2)
{\bf{Sum over r ((\textasciicircum{}n\textasciicircum{}C\textasciitilde{}r\textasciitilde{})\textasciicircum{}2\textasciicircum{}) = \textasciicircum{}2n\textasciicircum{}C\textasciitilde{}n\textasciitilde{} where r=0,...,n}} \newline % Row Count 16 (+ 2)
{\bf{Sum over s (\textasciicircum{}s\textasciicircum{}C\textasciitilde{}m\textasciitilde{}) = \textasciicircum{}(n+1)C\textasciitilde{}(m+1) where s=m,..,n}} \newline % Row Count 18 (+ 2)
{\bf{Hypergeometric expansion: \textasciicircum{}(n+m)\textasciicircum{}C\textasciitilde{}r\textasciitilde{} = \textasciicircum{}n\textasciicircum{}C\textasciitilde{}0\textasciitilde{}\textasciicircum{}m\textasciicircum{}C\textasciitilde{}r\textasciitilde{} + \textasciicircum{}n\textasciicircum{}C\textasciitilde{}1\textasciitilde{}\textasciicircum{}m\textasciicircum{}C\textasciitilde{}(r-1)\textasciitilde{} + ... + \textasciicircum{}n\textasciicircum{}C\textasciitilde{}r\textasciitilde{}\textasciicircum{}m\textasciicircum{}C\textasciitilde{}0\textasciitilde{} a CE and ME enumeration}} \newline % Row Count 21 (+ 3)
{\bf{n! = (n/e)\textasciicircum{}n\textasciicircum{} x root(2n x pi) - Stirling's approx. for n! }} \newline % Row Count 23 (+ 2)
{\bf{Trinomial expansion: (a+b+c)\textasciicircum{}n\textasciicircum{} = sum over i, j, k (C'a\textasciicircum{}i\textasciicircum{}b\textasciicircum{}j\textasciicircum{}c\textasciicircum{}k\textasciicircum{}) where i, j, k=0,...n and i+j+k=n and C'=n!/(i!j!k!)}} \newline % Row Count 26 (+ 3)
{\bf{In n-nomial expansion (a\textasciitilde{}1\textasciitilde{} + ... + a\textasciitilde{}r\textasciitilde{})\textasciicircum{}n\textasciicircum{} , the \# of terms in the sum is \textasciicircum{}r\textasciicircum{}\textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{}\textasciitilde{}n\textasciitilde{} = \textasciicircum{}(r+n-1)\textasciicircum{}C\textasciitilde{}n\textasciitilde{} = \textasciicircum{}(r+n-1)\textasciicircum{}C\textasciitilde{}(r-1)\textasciitilde{}}}% Row Count 29 (+ 3)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Expectation, Variance, RV Functions}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Expected value of X, ie: EX/E{[}X{]}/mu\textasciitilde{}X\textasciitilde{} = sum over all x\textasciitilde{}k\textasciitilde{} \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R\textasciitilde{}X\textasciitilde{} (x\textasciitilde{}k\textasciitilde{} . P(X = x\textasciitilde{}k\textasciitilde{})). It is linear}} \newline % Row Count 3 (+ 3)
{\bf{E{[}aX + b{]} = aE{[}X{]} + b, \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} a,b \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R}} \newline % Row Count 4 (+ 1)
{\bf{E{[}X\textasciitilde{}1\textasciitilde{} + ... + X\textasciitilde{}n\textasciitilde{}{]} = E{[}X\textasciitilde{}1\textasciitilde{}{]} + ... + E{[}X\textasciitilde{}n\textasciitilde{}{]}}} \newline % Row Count 6 (+ 2)
{\bf{If X is an RV and Y=g(X), then Y is also an RV.}} \newline % Row Count 8 (+ 2)
{\bf{R\textasciitilde{}Y\textasciitilde{} = \{g(x) | x \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R\textasciitilde{}X\textasciitilde{}\}}} and {\bf{P\textasciitilde{}Y\textasciitilde{}(y) = sum over all x:g(x)=y (P\textasciitilde{}X\textasciitilde{}(x))}} \newline % Row Count 10 (+ 2)
{\bf{E{[}g(X){]} = sum over all x\textasciitilde{}k\textasciitilde{} \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R\textasciitilde{}X\textasciitilde{} (g(x\textasciitilde{}k\textasciitilde{}) . P\textasciitilde{}X\textasciitilde{}(x\textasciitilde{}k\textasciitilde{})) (LOTUS)}} \newline % Row Count 12 (+ 2)
{\bf{Var(X) = E{[}(X - mu\textasciitilde{}X\textasciitilde{})\textasciicircum{}2\textasciicircum{}{]} = sum over all x\textasciitilde{}k\textasciitilde{} \textasciitilde{}\textasciitilde{}C\textasciitilde{}\textasciitilde{} R\textasciitilde{}X\textasciitilde{} ((x\textasciitilde{}k\textasciitilde{} - mu\textasciitilde{}X\textasciitilde{})\textasciicircum{}2\textasciicircum{}.P\textasciitilde{}X\textasciitilde{}(x\textasciitilde{}k\textasciitilde{}))}} \newline % Row Count 14 (+ 2)
{\bf{SD(X)/sigma\textasciitilde{}X\textasciitilde{} = sqrt(Var(X))}} \newline % Row Count 15 (+ 1)
{\bf{Covariance = Cov(X,Y) = E{[}XY{]} - E{[}X{]}.E{[}Y{]}, which will be 0 if X \& Y are independent}} \newline % Row Count 17 (+ 2)
{\bf{Var(X) = Cov (X,X) = E{[}X\textasciicircum{}2\textasciicircum{}{]} - (E{[}X{]})\textasciicircum{}2\textasciicircum{}}} \newline % Row Count 18 (+ 1)
{\bf{Var(aX + b) = a\textasciicircum{}2\textasciicircum{}Var(X), and if X = X\textasciitilde{}1\textasciitilde{} + ... + X\textasciitilde{}n\textasciitilde{}, then Var(X) = Var(X\textasciitilde{}1\textasciitilde{}) + ... + Var(X\textasciitilde{}n\textasciitilde{})}} \newline % Row Count 21 (+ 3)
{\bf{Var(aX + bY) = a\textasciicircum{}2\textasciicircum{}Var(X) + b\textasciicircum{}2\textasciicircum{}Var(Y) + 2abCov(X,Y)}} \newline % Row Count 23 (+ 2)
{\bf{Var(total up to X\textasciitilde{}n\textasciitilde{}) = sum of all Var(X\textasciitilde{}i\textasciitilde{}) if X\textasciitilde{}i\textasciitilde{} is mutually independent for i = 1...n. Summing up over the same conditions for expected values holds true, regardless of independence or not}} \newline % Row Count 27 (+ 4)
{\bf{Correlation coefficient = Cov(X,Y)/(SD(X) . SD(Y)) - ranges between -1 and 1 (inclusive for both limits)}} \newline % Row Count 30 (+ 3)
} \tn 
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Expectation, Variance, RV Functions (cont)}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Z-standardized transformation: Z=(X - mu\textasciitilde{}X\textasciitilde{})/SD(X) - zero mean and unit variance}}% Row Count 2 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


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

\end{document}