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='''1.2.1 Probability Space'''= | ='''1.2.1 Probability Space'''= | ||
| − | • Probability Space = <math>\left\{ \mathcal{S},\mathcal{F}\left(\mathcal{S}\right),\mathcal{P}\right\}</math> | + | • Probability Space = <math class="inline">\left\{ \mathcal{S},\mathcal{F}\left(\mathcal{S}\right),\mathcal{P}\right\}</math> |
| − | • <math>\mathcal{S}\sim</math> sample space | + | • <math class="inline">\mathcal{S}\sim</math> sample space |
| − | • <math>\mathcal{F}\left(\mathcal{S}\right)\sim</math> event space , collection of subsets of <math>\mathcal{S}</math> (including sample space itself) | + | • <math class="inline">\mathcal{F}\left(\mathcal{S}\right)\sim</math> event space , collection of subsets of <math class="inline">\mathcal{S}</math> (including sample space itself) |
| − | • <math>\mathcal{P}\sim</math> maps <math>\mathcal{F}\left(\mathcal{S}\right)\rightarrow\left[0,1\right]</math> | + | • <math class="inline">\mathcal{P}\sim</math> maps <math class="inline">\mathcal{F}\left(\mathcal{S}\right)\rightarrow\left[0,1\right]</math> |
='''1.2.2 Event space'''= | ='''1.2.2 Event space'''= | ||
| − | • Event space <math>F\left(S\right)</math> or <math>F</math> is a non-empty collection of subset of <math>S</math> satisfying the following three closure properties: | + | • Event space <math class="inline">F\left(S\right)</math> or <math class="inline">F</math> is a non-empty collection of subset of <math class="inline">S</math> satisfying the following three closure properties: |
| − | 1. If <math>A\in F\left(S\right)</math> , then <math>\bar{A}\in F\left(S\right)</math> . | + | 1. If <math class="inline">A\in F\left(S\right)</math> , then <math class="inline">\bar{A}\in F\left(S\right)</math> . |
| − | 2. If for some finite <math>n</math> , <math>A_{1},A_{2},\cdots,A_{n}\in F\left(S\right)</math> , then <math>\bigcup_{i=1}^{n}A_{i}\in F\left(S\right)</math> . | + | 2. If for some finite <math class="inline">n</math> , <math class="inline">A_{1},A_{2},\cdots,A_{n}\in F\left(S\right)</math> , then <math class="inline">\bigcup_{i=1}^{n}A_{i}\in F\left(S\right)</math> . |
| − | 3. If <math>A_{i}\in F\left(S\right)</math> , <math>i=1,2,\cdots</math> , then <math>\bigcup_{i=1}^{\infty}A_{i}\in F\left(S\right)</math> . | + | 3. If <math class="inline">A_{i}\in F\left(S\right)</math> , <math class="inline">i=1,2,\cdots</math> , then <math class="inline">\bigcup_{i=1}^{\infty}A_{i}\in F\left(S\right)</math> . |
| − | • A set <math>F\left(S\right)</math> with these 3 properties is called a sigma-field (<math>\sigma</math>-field). If only 1 and 2 are satisfied, we have a field. | + | • A set <math class="inline">F\left(S\right)</math> with these 3 properties is called a sigma-field (<math class="inline">\sigma</math>-field). If only 1 and 2 are satisfied, we have a field. |
| − | • It follows from three properties that <math>\varnothing,S\in F\left(S\right)</math> . | + | • It follows from three properties that <math class="inline">\varnothing,S\in F\left(S\right)</math> . |
| − | – Suppose <math>A\in F\left(S\right)</math> , then <math>\bar{A}\in F\left(S\right)</math> , <math>A\cup\bar{A}=S\in F\left(S\right)</math> , and <math>\bar{S}=\varnothing\in F\left(S\right)</math> . | + | – Suppose <math class="inline">A\in F\left(S\right)</math> , then <math class="inline">\bar{A}\in F\left(S\right)</math> , <math class="inline">A\cup\bar{A}=S\in F\left(S\right)</math> , and <math class="inline">\bar{S}=\varnothing\in F\left(S\right)</math> . |
| − | • What about intersection? Suppose <math>A,B\in F\left(S\right)</math> . Is <math>A\cap B\in F\left(S\right)</math> ? | + | • What about intersection? Suppose <math class="inline">A,B\in F\left(S\right)</math> . Is <math class="inline">A\cap B\in F\left(S\right)</math> ? |
| − | – <math>A\cap B=\overline{\overline{A\cap B}}=\overline{\overline{A}\cup\overline{B}}\in F\left(S\right)</math> . | + | – <math class="inline">A\cap B=\overline{\overline{A\cap B}}=\overline{\overline{A}\cup\overline{B}}\in F\left(S\right)</math> . |
='''1.2.3 Axioms of probability'''= | ='''1.2.3 Axioms of probability'''= | ||
| − | • The probability measure <math>P\left(\cdot\right)</math> corresponding to <math>S</math> and <math>F\left(S\right)</math> is the assignment of a real number <math>P\left(A\right)</math> to each <math>A\in F\left(S\right)</math> satisfying following properties. Axioms of probability: | + | • The probability measure <math class="inline">P\left(\cdot\right)</math> corresponding to <math class="inline">S</math> and <math class="inline">F\left(S\right)</math> is the assignment of a real number <math class="inline">P\left(A\right)</math> to each <math class="inline">A\in F\left(S\right)</math> satisfying following properties. Axioms of probability: |
| − | 1. <math>P\left(A\right)\geq0</math> , <math>\forall A\in F\left(S\right)</math> . | + | 1. <math class="inline">P\left(A\right)\geq0</math> , <math class="inline">\forall A\in F\left(S\right)</math> . |
| − | 2. <math>P\left(S\right)=1</math> . | + | 2. <math class="inline">P\left(S\right)=1</math> . |
| − | 3. If <math>A_{1}</math> and <math>A_{2}</math> are disjoint events, then <math>P\left(A_{1}\cup A_{2}\right)=P\left(A_{1}\right)+P\left(A_{2}\right)</math> . If <math>A_{1},A_{2}\in F\left(S\right)</math> and <math>A_{1}\cap A_{2}=\varnothing</math> , then <math>A_{1}</math> and <math>A_{2}</math> are disjoint events. | + | 3. If <math class="inline">A_{1}</math> and <math class="inline">A_{2}</math> are disjoint events, then <math class="inline">P\left(A_{1}\cup A_{2}\right)=P\left(A_{1}\right)+P\left(A_{2}\right)</math> . If <math class="inline">A_{1},A_{2}\in F\left(S\right)</math> and <math class="inline">A_{1}\cap A_{2}=\varnothing</math> , then <math class="inline">A_{1}</math> and <math class="inline">A_{2}</math> are disjoint events. |
| − | 4. If <math>A_{1},A_{2},\cdots,A_{n},\cdots\in F\left(S\right)</math> is a countable collection of disjointed events, then <math>P\left(\bigcup_{i=1}^{\infty}A_{i}\right)=\sum_{i=1}^{\infty}P\left(A_{i}\right)</math> . | + | 4. If <math class="inline">A_{1},A_{2},\cdots,A_{n},\cdots\in F\left(S\right)</math> is a countable collection of disjointed events, then <math class="inline">P\left(\bigcup_{i=1}^{\infty}A_{i}\right)=\sum_{i=1}^{\infty}P\left(A_{i}\right)</math> . |
| − | • <math>P\left(\cdot\right)</math> is a set function. <math>P\left(\cdot\right):F\left(S\right)\rightarrow\mathbf{R}</math> . | + | • <math class="inline">P\left(\cdot\right)</math> is a set function. <math class="inline">P\left(\cdot\right):F\left(S\right)\rightarrow\mathbf{R}</math> . |
| − | • If you want to talk about the probability of a single output <math>\omega_{0}\in S</math> , you do so by considering the single event | + | • If you want to talk about the probability of a single output <math class="inline">\omega_{0}\in S</math> , you do so by considering the single event |
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Latest revision as of 10:21, 30 November 2010
Contents
1.2 Probability Space
From the ECE600 Pre-requisites notes of Sangchun Han, ECE PhD student.
1.2.1 Probability Space
• Probability Space = $ \left\{ \mathcal{S},\mathcal{F}\left(\mathcal{S}\right),\mathcal{P}\right\} $
• $ \mathcal{S}\sim $ sample space
• $ \mathcal{F}\left(\mathcal{S}\right)\sim $ event space , collection of subsets of $ \mathcal{S} $ (including sample space itself)
• $ \mathcal{P}\sim $ maps $ \mathcal{F}\left(\mathcal{S}\right)\rightarrow\left[0,1\right] $
1.2.2 Event space
• Event space $ F\left(S\right) $ or $ F $ is a non-empty collection of subset of $ S $ satisfying the following three closure properties:
1. If $ A\in F\left(S\right) $ , then $ \bar{A}\in F\left(S\right) $ .
2. If for some finite $ n $ , $ A_{1},A_{2},\cdots,A_{n}\in F\left(S\right) $ , then $ \bigcup_{i=1}^{n}A_{i}\in F\left(S\right) $ .
3. If $ A_{i}\in F\left(S\right) $ , $ i=1,2,\cdots $ , then $ \bigcup_{i=1}^{\infty}A_{i}\in F\left(S\right) $ .
• A set $ F\left(S\right) $ with these 3 properties is called a sigma-field ($ \sigma $-field). If only 1 and 2 are satisfied, we have a field.
• It follows from three properties that $ \varnothing,S\in F\left(S\right) $ .
– Suppose $ A\in F\left(S\right) $ , then $ \bar{A}\in F\left(S\right) $ , $ A\cup\bar{A}=S\in F\left(S\right) $ , and $ \bar{S}=\varnothing\in F\left(S\right) $ .
• What about intersection? Suppose $ A,B\in F\left(S\right) $ . Is $ A\cap B\in F\left(S\right) $ ?
– $ A\cap B=\overline{\overline{A\cap B}}=\overline{\overline{A}\cup\overline{B}}\in F\left(S\right) $ .
1.2.3 Axioms of probability
• The probability measure $ P\left(\cdot\right) $ corresponding to $ S $ and $ F\left(S\right) $ is the assignment of a real number $ P\left(A\right) $ to each $ A\in F\left(S\right) $ satisfying following properties. Axioms of probability:
1. $ P\left(A\right)\geq0 $ , $ \forall A\in F\left(S\right) $ .
2. $ P\left(S\right)=1 $ .
3. If $ A_{1} $ and $ A_{2} $ are disjoint events, then $ P\left(A_{1}\cup A_{2}\right)=P\left(A_{1}\right)+P\left(A_{2}\right) $ . If $ A_{1},A_{2}\in F\left(S\right) $ and $ A_{1}\cap A_{2}=\varnothing $ , then $ A_{1} $ and $ A_{2} $ are disjoint events.
4. If $ A_{1},A_{2},\cdots,A_{n},\cdots\in F\left(S\right) $ is a countable collection of disjointed events, then $ P\left(\bigcup_{i=1}^{\infty}A_{i}\right)=\sum_{i=1}^{\infty}P\left(A_{i}\right) $ .
• $ P\left(\cdot\right) $ is a set function. $ P\left(\cdot\right):F\left(S\right)\rightarrow\mathbf{R} $ .
• If you want to talk about the probability of a single output $ \omega_{0}\in S $ , you do so by considering the single event
