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% Document Info
\author{Dylan (dylablo)}
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  /Title (foundation-of-statistics-sec-3-under-shirin.pdf)
  /Creator (Cheatography)
  /Author (Dylan (dylablo))
  /Subject (Foundation of Statistics Sec. 3 under Shirin Cheat Sheet)
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    {\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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    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Foundation of Statistics Sec. 3 under Shirin Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{Dylan (dylablo)} via \textcolor{DarkBackground}{\uline{cheatography.com/68322/cs/17264/}}}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}Dylan (dylablo) \\
  \uline{cheatography.com/dylablo} \\
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 1st October, 2018.\\
   Page {\thepage} of \pageref{LastPage}.
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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
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\begin{multicols*}{2}

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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Binomial \& Bernoulli Distributions}}  \tn
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\mymulticolumn{1}{x{8.4cm}}{Introduction:\{\{nl\}\}Given an experiment with 2 possible outcomes (Success \& Failure ie. 1 and 0 ie. Binary) ran 5 times.\{\{nl\}\}Sample space Ω becomes a combination of the 5 results ie. Ω = \{0,1\} x \{0,1\} x \{0,1\} x \{0,1\} x \{0,1\}.\{\{nl\}\}With the information that event A is when any one experiment is a success we are able to deduce that set A = \{(0,0,0,0,1),(0,0,0,1,0),(0,0,1,0,0),(0,1,0,0,0),(1,0,0,0,0)\}\{\{nl\}\}Equate the probability of success happening (not necessarily in A, only in general as a result of the experiment being ran any 1 time) is given by p, and consequently (as there are only 2 possibilities) failure is given by 1-p. Remember p is the probability of a single result occuring that is considered a success. In the case we are flipping a coin and heads is a success p = P(H) = 1/2, in the case of a dice where 5 is considered a success p = P(5) = 1/6\{\{nl\}\}We can obviously see that P(A) = 5(1-p)\textasciicircum{}4\textasciicircum{}p or in plain english - "There are 5 possible combinations of A occuring, contained within each 4 failures (1-p)\textasciicircum{}4\textasciicircum{} and 1 success p"} \tn 
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\mymulticolumn{1}{x{8.4cm}}{Bernoulli Distribution...\{\{nl\}\}- Used to model experiements with binary outcomes (success or failure)\{\{nl\}\}- Definition: Discrete rv X has a Bernoulli distribution with parameter p (p is the probability of success occuring in any given experiment) where 0\textless{}=p\textless{}=1 if its probability mass function (pmf) is given by...\{\{nl\}\}px(1) = P(X=1) = p\{\{nl\}\}and\{\{nl\}\}px(0) = P(x=0) = 1-p\{\{nl\}\}ie. Probability of success occuring is p and probability of failure is 1-p} \tn 
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Binomial \& Bernoulli Distributions (cont)}}  \tn
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\mymulticolumn{1}{x{8.4cm}}{Binomial Distribution:\{\{nl\}\}A discrete rv X has a binomial distribution with parameters n and p where n = 1,2... and 0\textless{}=p\textless{}=1, if the pmf is given by... Px(k) = P(X = k) = (`k`\textasciicircum{}n\textasciicircum{})p\textasciicircum{}k\textasciicircum{}(1-p)\textasciicircum{}n-k\textasciicircum{}\{\{nl\}\}\{\{nl\}\}Denoted by Bin(n,p)\{\{nl\}\}Expectation of binomial distribution Bin(n,p) is E(X)=np\{\{nl\}\}Variance is Var(X)=np(1-p)} \tn 
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Hypergeometric Random Variable}}  \tn
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\mymulticolumn{1}{x{8.4cm}}{Introduction:\{\{nl\}\}When an experiement consists of...\{\{nl\}\}1. Drawing n random elements WITHOUT REPLACEMENT from a set of N elements.\{\{nl\}\}2. s elements of N are special.\{\{nl\}\}3. N - s are regular\{\{nl\}\}Result: Our Hypergeometric rv X is the number of special elements in our random draw of n ie s∩n (sortof)} \tn 
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{The Geometric Distribution}}  \tn
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\mymulticolumn{1}{x{8.4cm}}{Introductin:\{\{nl\}\}1. Observations have 2 possible values Success and Failures\{\{nl{]}\}2. Probability of success (p) is the same for each observation.\{\{nl\}\}Observations are all independent.\{\{nl\}\}We are looking for the number of trials required to find the first success.\{\{nl\}\}\{\{nl\}\}This is an example of an infinite experiment...} \tn 
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\mymulticolumn{1}{x{8.4cm}}{A discrete rv X has a geometric distribution with parameter p, where 0\textless{}=p\textless{}=1, if its pmf is given by \{\{nl\}\}px(k) = P(X=k) = (1-p)\textasciicircum{}k-1\textasciicircum{}p\{\{nl\}\}for k=1,2,....\{\{nl\}\}This distribution is denoted by Geo(p)\{\{nl\}E(X) = Σ`k=1`\textasciicircum{}∞\textasciicircum{}kp(1-p)\textasciicircum{}k-1\textasciicircum{}=1/p\{\{nl\}\}Var(X) = (1-p)/p\textasciicircum{}2\textasciicircum{}} \tn 
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\mymulticolumn{1}{x{8.4cm}}{Memoryless Property:\{\{nl\}\}P(X\textgreater{}n+k|X\textgreater{}k) = P(X\textgreater{}n)} \tn 
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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Poisson Distribution}}  \tn
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\mymulticolumn{1}{x{8.4cm}}{Introduction:\{\{nl\}\}Used when we are interested in the number of X per the number of Y. eg. The number of cars on a road per hour. This is phrase as counts per unit interval.\{\{nl\}\}When count is relatively low it can be modelled as a Poisson Distribution.} \tn 
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\mymulticolumn{1}{x{8.4cm}}{Assumptions:\{\{n\}\}1. Homogeneity: The rate λ (the rate at which the event occurs) is constant over time/space. This implies that at any unit interval E(X)=λ\{\{nl\}\}2. Independence: Number of events in disjoint intervals are independent of one another.} \tn 
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\mymulticolumn{1}{x{8.4cm}}{Please refer to Definitions and Other Formulae sheet for more information.} \tn 
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\mymulticolumn{1}{x{8.4cm}}{Property of Poisson:\{\{nl\}\} If we sum 2 Poisson rv X and Y, our result is also Poisson.\{\{nl\}x\textasciitilde{}Poisson(λ) and Y\textasciitilde{}Poisson(µ) then X+Y\textasciitilde{}Poisson(λ+µ)} \tn 
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% That's all folks
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