Symbols:P

From ProofWiki
Jump to navigation Jump to search

Previous  ... Next

pico-

$\mathrm p$

The Système Internationale d'Unités symbol for the metric scaling prefix pico, denoting $10^{\, -12 }$, is $\mathrm { p }$.


Its $\LaTeX$ code is \mathrm {p} .


peta-

$\mathrm P$

The Système Internationale d'Unités symbol for the metric scaling prefix peta, denoting $10^{\, 15 }$, is $\mathrm { P }$.


Its $\LaTeX$ code is \mathrm {P} .


Prime Number

$p$

Used to denote a general prime number.


The $\LaTeX$ code for \(p\) is p .


Probability

$p$

Used to denote a general probability.

As such, $p$ is a real number such that:

$0 \le p \le 1$


The $\LaTeX$ code for \(p\) is p .


Power Set

$\powerset S$ is the power set of the set $S$.

It is defined as:

$\powerset S = \set {T: T \subseteq S}$


$\map {\mathfrak P} S$ is an alternative notation, but the fraktur font, of which $\mathfrak P$ is an example, is falling out of use, probably as a result of its difficulty in being both read and written.


The $\LaTeX$ code for \(\powerset S\) is \powerset S .

The $\LaTeX$ code for \(\map {\mathfrak P} S\) is \map {\mathfrak P} S .


Poisson Distribution

$X \sim \Poisson \lambda$

or

$X \sim \map {\operatorname {Pois} } \lambda$


$X$ has the Poisson distribution with parameter $\lambda$.


The $\LaTeX$ code for \(X \sim \Poisson \lambda\) is X \sim \Poisson \lambda .

The $\LaTeX$ code for \(X \sim \map {\operatorname {Pois} } \lambda\) is X \sim \map {\operatorname {Pois} } \lambda .


Probability Measure

$\Pr$


Let $\EE$ be an experiment.

Definition 1

Let $\EE$ be defined as a measure space $\struct {\Omega, \Sigma, \Pr}$.

Then $\Pr$ is a measure on $\EE$ such that $\map \Pr \Omega = 1$.


Definition 2

Let $\Omega$ be the sample space on $\EE$.

Let $\Sigma$ be the event space of $\mathcal E$.


A probability measure on $\EE$ is a mapping $\Pr: \Sigma \to \R$ which fulfils the Kolmogorov axioms:


\((1)\)   $:$     \(\displaystyle \forall A \in \Sigma:\)    \(\displaystyle 0 \)   \(\displaystyle \le \)   \(\displaystyle \map \Pr A \le 1 \)             The probability of an event occurring is a real number between $0$ and $1$
\((2)\)   $:$      \(\displaystyle \map \Pr \Omega \)   \(\displaystyle = \)   \(\displaystyle 1 \)             The probability of some elementary event occurring in the sample space is $1$
\((3)\)   $:$      \(\displaystyle \map \Pr {\bigcup_{i \mathop \ge 1} A_i} \)   \(\displaystyle = \)   \(\displaystyle \sum_{i \mathop \ge 1} \map \Pr {A_i} \)             where $\set {A_1, A_2, \ldots}$ is a countable (possibly countably infinite) set of pairwise disjoint events
         That is, the probability of any one of countably many pairwise disjoint events occurring is the sum of the probabilities of the occurrence of each of the individual events


Definition 3

Let $\Omega$ be the sample space on $\EE$.

Let $\Sigma$ be the event space of $\mathcal E$.


A probability measure on $\EE$ is a mapping $\Pr: \Sigma \to \R$ which fulfils the following axioms:


\((\text {I})\)   $:$     \(\displaystyle \forall A \in \Sigma:\)    \(\displaystyle \map \Pr A \)   \(\displaystyle \ge \)   \(\displaystyle 0 \)             
\((\text {II})\)   $:$      \(\displaystyle \map \Pr \Omega \)   \(\displaystyle = \)   \(\displaystyle 1 \)             
\((\text {III})\)   $:$     \(\displaystyle \forall A \in \Sigma:\)    \(\displaystyle \map \Pr A \)   \(\displaystyle = \)   \(\displaystyle \sum_{\bigcup \set e \mathop = A} \map \Pr {\set e} \)             where $e$ denotes the elementary events of $\EE$


Definition 4

Let $\Omega$ be the sample space on $\EE$.

Let $\Sigma$ be the event space of $\mathcal E$.


A probability measure on $\EE$ is an additive function $\Pr: \Sigma \to \R$ which fulfils the following axioms:


\((1)\)   $:$     \(\displaystyle \forall A, B \in \Sigma: A \cap B = \O:\)    \(\displaystyle \map \Pr {A \cup B} \)   \(\displaystyle = \)   \(\displaystyle \map \Pr A + \map \Pr B \)             
\((2)\)   $:$      \(\displaystyle \map \Pr \Omega \)   \(\displaystyle = \)   \(\displaystyle 1 \)             


The $\LaTeX$ code for \(\Pr\) is \Pr .


Probability Mass Function

$\map {p_X} x$


Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X: \Omega \to \R$ be a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.


Then the (probability) mass function of $X$ is the (real-valued) function $p_X: \R \to \closedint 0 1$ defined as:

$\forall x \in \R: \map {p_X} x = \begin{cases} \map \Pr {\set {\omega \in \Omega: \map X \omega = x} } & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end{cases}$

where $\Omega_X$ is defined as $\Img X$, the image of $X$.

That is, $\map {p_X} x$ is the probability that the discrete random variable $X$ takes the value $x$.


The $\LaTeX$ code for \(\map {p_X} x\) is \map {p_X} x .