Lecture 1. A) Sample Space

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Sample Space


Traditionally, in probability theory, tossing a coin can yield only Heads or Tails. The set of all possible cases, Heads or Tails in this case, is called the sample space. It’s the space from which one can sample from, for example, or the space from which a realization can be produced from an experiment. We denote the sample space by [math]S.[/math] For example, in the coin tossing example, [math]S=\left\{Heads,Tails\right\}.[/math]

Let's introduce some more notation:
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  • [math]\emptyset[/math] is the empty set. It can be denoted as [math]\emptyset=\left\{\right\}.[/math]

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  • [math]\bigcup_{i=1}^{\infty}B_{i}[/math] is the union of sets [math]B_{i}.[/math] Formally, [math]\bigcup_{i=1}^{\infty}B_{i}=\left\{ s\in S:s\in B_{i}\,for\,some\,i\right\}.[/math]

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  • [math]B\subseteq S[/math] means [math]B[/math] is a subset of the sample space. For example, [math]B=\left\{ Heads\right\}[/math] is a subset of [math]S=\left\{ Heads,Tails\right\}.[/math]

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  • [math]Heads[/math], without curly braces, is an element of set [math]B.[/math]

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  • [math]B^{C}=S\backslash B[/math] is the complement of set [math]B[/math]. When [math]B=\left\{ Heads\right\}[/math], [math]B^{C}=\left\{ Tails\right\}.[/math] Formally, [math]B^{C}=\left\{ s\in S:s\notin B\right\}.[/math]