# Symbols:P

## Contents

## 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}`

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## 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}`

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## Prime Number

- $p$

Used to denote a general prime number.

The $\LaTeX$ code for \(p\) is `p`

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## 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`

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## 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`

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

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## 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`

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

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## 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`

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## 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`

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