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% Document Info
\author{Delegado FM (Learningbizz)}
\pdfinfo{
  /Title (mve137-chalmers-university.pdf)
  /Creator (Cheatography)
  /Author (Delegado FM (Learningbizz))
  /Subject (MVE137 - Chalmers University Cheat Sheet)
}

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    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{MVE137 - Chalmers University Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{Delegado FM (Learningbizz)} via \textcolor{DarkBackground}{\uline{cheatography.com/73767/cs/34116/}}}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}Delegado FM (Learningbizz) \\
  \uline{cheatography.com/learningbizz} \\
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 25th September, 2022.\\
   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{2.4885 cm} x{2.4885 cm} }
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Basic Probability Definitions}}  \tn
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Sample Space (Ω) & Set of all possible outcomes of a random experiment. \tn 
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Event & Outcome of a random experiment (inside Ω) \tn 
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σ-field & The allowable events constitute a family of sets F, usually referred to as σ-field. Each set in F is a subset of the sample space Ω. \tn 
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Probability measure (P) & A probability measure on (Ω, F) is a function P : F → {[}0, 1{]} that satisfies the following two properties: \{\{nl\}\} 1. P{[}Ω{]} = 1  \{\{nl\}\} 2. The probability of the union of a collection of disjoint members is the sum of its probabilities \tn 
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Probability space & (Ω, F, P) \tn 
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Basic properties of probability measures & P{[}∅{]} = 0 \{\{nl\}\} P{[}A¯{]} = 1 − P{[}A{]} \{\{nl\}\} If A ⊂ B, then P{[}B{]} = P{[}A{]} + P{[}B \textbackslash{} A{]} ≥ P{[}A{]} \{\{nl\}\} P{[}A ∪ B{]} = P{[}A{]} + P{[}B{]} − P{[}A ∩ B{]} \tn 
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Basic Probability Definitions (cont)}}  \tn
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Inclusion Exclusion Principle \{\{nl\}\} (comes from last basic property of probability measures) & Given sets A1, A2... \{\{nl\}\} {\bf{P{[}union(Ai){]} ≤ sum(P{[}Ai{]})}} \{\{nl\}\} When the two events are disjoint, the inequality is = as they don't share any common space: P{[}A ∩ B{]} = 0 \tn 
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Sampling strategy & Choose repeatedly a random number in Ω \tn 
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Sampling with replacement & Select random numbers in Ω, {\bf{without taking into account}} which ones you've already tested. Therefore, there will be some numbers tested multiple times \tn 
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Sampling without replacement & Select random numbers in Ω {\bf{taking into account}} which ones you've already used. Therefore, you won't run the algorithm with the same number more than once \tn 
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Independent (events or family) & Two events are independent if: \{\{nl\}\} P{[}A ∩ B{]} = P{[}A{]} P{[}B{]} \{\{nl\}\} It also applies to families \{Ai, i∈ I\} \tn 
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Basic Probability Definitions (cont)}}  \tn
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Pairwise & To form all possible pairs (two items at a time) from a set \tn 
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Pairwise independent (family or events) & A family or events are pairwise independent if: \{\{nl\}\} {\bf{P{[}Ai ∩ Aj {]} = P{[}Ai{]} P{[}Aj{]} for all i  != j}} \{\{nl\}\} In english terms, a family or events is pairwise independent if any of its possible pairs is independent of each other. For example: \{\{nl\}\} P(A∩B)=P(A)P(B) \{\{nl\}\}  P(A∩C)=P(A)P(C) \{\{nl\}\} P(B∩C)=P(B)P(C) \tn 
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Mutually independent (events) & More than two events (i.e. A,B,C) are mutually independent if: \{\{nl\}\} 1. They are pairwise independent \{\{nl\}\} 2. They meet the condition: \{\{nl\}\} P(A ∩ B ∩ C) = P(A) × P(B) × P(C) \{\{nl\}\} In plain english, events are mutually independent if any event is independent to the other events \tn 
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Conditional Probability & If P{[}B{]} \textgreater{} 0, the conditional probability that A occurs give that B occurs is: P{[}A|B{]}=P{[}A∩B{]}/P{[}B{]} \tn 
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Conditional Probability (independent events) & If A and B are independent events, then: \{\{nl\}\} P{[}A|B{]} = P{[}A∩ B{]}/P{[}B{]} = \{\{nl\}\}(P{[}A{]}*P{[}B{]})/P{[}B{]} = P{[}A{]} \tn 
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Law of Total Probability & Let e1...en be {\bf{partitions}} of Ω\{\{nl\}\}(a collection of ALL the sets in Ω which are independent of each other). Also assuming P{[}ei{]} \textgreater{} 0 for all i. The probability of A can be written as: \{\{nl\}\} P{[}A{]} = sum(i=1,n)(P{[}A|ei{]}*P{[}ei{]})\{\{nl\}\} In english, it's the sum of all the possible scenarios in which A can occur \tn 
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Bayes Theorem & Assuming e1...en be {\bf{partitions}} of Ω: \{\{nl\}\} P{[}ej|B{]} = P{[}Ej ∩ B{]}/P{[}B{]} = (P{[}B|Ej{]}{\emph{P{[}Ej{]})/(sum(i=1,n)(P{[}B|ei{]}}}P{[}ei{]}) \{\{nl\}\} It's basically using conditional theory and then applying conditional theory again for the top part and law of total probability in the lower part \tn 
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Discrete Random Variables and Expectation}}  \tn
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Random Variable & A random variable X on a sample space Ω is a real-valued (measurable) function on Ω; that is X : Ω → R.\{\{nl\}\}Denoted as upper case in this course and real numbers as lower case \tn 
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Discrete Random Variable & A discrete random variable is a random variable that outputs only a finite or countably infinite number of values\{\{nl\}\}(i.e. number of kids in a family, range between 1 and x) \tn 
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Probability that X=a & Sum of all the events {\emph{w}} in Ω which X(w) = x \tn 
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Independence of random variables & Two random variables X and Y are independent if and only if: \{\{nl\}\}P{[}(X = x)∩(Y = y){]} = P{[}X=x{]}*P{[}Y=y{]} \{\{nl\}\}for all values x and y \tn 
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Mutually independent random variables & Like mutually independent events \tn 
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Discrete Random Variables and Expectation (cont)}}  \tn
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Expectation (mean) & It is a weighted average of the values assumed by the random variable, taking into account the probability of getting that value.\{\{nl\}\}The expectation of a discrete random variable X, denoted by E{[}X{]} is given by\{\{nl\}\}E{[}X{]} = sum(i=x,X)(x*P{[}X = x{]}) \tn 
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% That's all folks
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