Flashcards
Probability and Measure
Cards in this deck (8)
- 1Pairwise DisjointA number of sets are pairwise disjoint if every pair of the sets are disjoint. Eg, the three sets \(\{1, 2\}, \{3, 4\}\) and \(\{5, 6\}\). Alternatively, one can ask for the weaker condition that the sets have empty intersection. For instance, the three sets \(\{1, 2\}, \{1, 3\}\) and \(\{2, 4\}\) have empty intersection but are not pairwise disjoint.
- 2Power SetThe power set (or powerset) of any set \( (S)\), written \(P(S)\), is the set of all subsets of \(S\), including the empty set and \(S\) itself.
- 3BilinearLet \(V, W\) and \(X\) be three vector spaces over the same base field \(F\). A bilinear map is a function
\[B : V × W → X\]
such that for any \(w \in W\) the map
\[v ↦ B(v, w)\]
is a linear map from \(V\) to \(X\), and for any \(v \in V\) the map
\[w ↦ B(v, w)\]
is a linear map from \(W\) to \(X\).
In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed. - 4What are the requirements for a function \(\mu\) to be a probability measure on a probability space?1. \(\mu\) must return results in the unit interval \([0, 1]\) returning \(0\) for the empty set and $1$ for the entire space.
2. \(\mu\) must satisfy the countable additivity property that for all countable collections \(\{E_i\}\) of pairwise disjoint sets:
\[\mu(\displaystyle\cup_{i=1}^\infty E_i) = \sum_{i=1}^\infty \mu(E_i).\] - 5MeasureLet \(X\) be a set and \(\Sigma\) a \(\sigma\)-algebra over \(X\). A function \(\mu\) from \(\Sigma\) to the extended real number line is called a measure if it satisfies the following properties:
1. Non-negativity: For all \(E \in \Sigma:\) \(\mu(E) \ge 0.\)
2. Null empty set: \(\mu(\emptyset) = 0\).
3. Countable additivity (or \(\sigma\)-additivity): For all countable collections \(E_n\) of pairwise disjoint sets in \(\Sigma\):
\[\mu(\displaystyle\cup_1^\infty E_n)=\sum_1^\infty \mu(E_n)\]
(4) At least one set \(E\) has finite measure. So that the empty set automatically has measure zero because of countable additivity:
\[\mu(E)=\mu\left(E\cup\emptyset\right) = \mu(E) + \mu(\emptyset)$, so $\mu(\emptyset)= \mu(E) - \mu(E) = 0.\] - 6Borel MeasureLet \(X\) be a locally compact Hausdorff space, and let \(\mathfrak{B}(X)\) be the smallest \(\sigma\)-algebra that contains the open sets of \(X\); this is known as the \(\sigma\)-algebra of Borel sets. A Borel measure is any measure \(\mu\) defined on the \(\sigma\)-algebra of Borel sets.
- 7Probability space - DiscreteDiscrete probability theory needs only at most countable sample spaces {\displaystyle \Omega } . Probabilities can be ascribed to points of {\displaystyle \Omega } by the probability mass function {\displaystyle p:\Omega \to [0,1]} such that {\displaystyle \textstyle \sum _{\omega \in \Omega }p(\omega )=1} . All subsets of {\displaystyle \Omega } can be treated as events (thus, {\displaystyle {\mathcal {F}}=2^{\Omega }} is the power set). The probability measure takes the simple form
{\displaystyle (*)\qquad P(A)=\sum _{\omega \in A}p(\omega )\quad {\text{for all }}A\subseteq \Omega .}
The greatest σ-algebra {\displaystyle {\mathcal {F}}=2^{\Omega }} describes the complete information. In general, a σ-algebra {\displaystyle {\mathcal {F}}\subseteq 2^{\Omega }} corresponds to a finite or countable partition {\displaystyle \Omega =B_{1}\cup B_{2}\cup \dots } , the general form of an event {\displaystyle A\in {\mathcal {F}}} being {\displaystyle A=B_{k_{1}}\cup B_{k_{2}}\cup \dots } . See also the examples.
The case {\displaystyle p(\omega )=0} is permitted by the definition, b - 8Probability SpaceA probability space is a triple \((\Omega ,{\mathcal {F}},P)\) consisting of:
1. The sample space \( \Omega\) - an arbitrary non-empty set
2. The σ-algebra \( {\mathcal {F}}\subseteq 2^{\Omega }\) (also called σ-field) - a set of subsets of \(\Omega\), called events, such that:
i) \(\Omega \in {\mathcal {F}}\),
ii) If \(A\in {\mathcal {F}}\) , then also \((\Omega \setminus A)\in {\mathcal {F}}\),
iii) If \(A_{i} \in {\mathcal {F}}\) for \(i=1,2,\dots \), then also \(\textstyle (\bigcup _{i=1}^{\infty }A_{i})\in {\mathcal {F}}\)
3. The probability measure \(P:{\mathcal {F}}\to [0,1]\) - a function on \({\mathcal {F}}\) such that:
i) if \(\{A_{i}\}_{i=1}^{\infty }\subseteq {\mathcal {F}}\) is a countable collection of pairwise disjoint sets, then \(\textstyle P(\bigcup _{i=1}^{\infty }A_{i})=\sum _{i=1}^{\infty }P(A_{i})\)
ii) \(P(\Omega )=1\).
Information
- Description
- PhD Mathematics (Free Probability) Flashcards on Probability and Measure, created by L H on 22/05/2016.
- Created by
- L H
- Views
- 2
- Rating