# Symbols:Abbreviations/P

## P

### PCI

### PCFI

### PDE

A **partial differential equation** (abbreviated **P.D.E.** or **PDE**) is a **differential equation** which has:

- one dependent variable
- more than one independent variable.

The derivatives occurring in it are therefore partial.

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

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

Let $\Omega_X = \Img X$, the image of $X$.

Then the **probability density function** of $X$ is the mapping $f_X: \R \to \closedint 0 1$ defined as:

- $\forall x \in \R: \map {f_X} x = \begin {cases} \displaystyle \lim_{\epsilon \mathop \to 0^+} \frac {\map \Pr {x - \frac \epsilon 2 \le X \le x + \frac \epsilon 2} } \epsilon & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end {cases}$

### PFI

### PGF or p.g.f.

Let $X$ be a discrete random variable whose codomain, $\Omega_X$, is a subset of the natural numbers $\N$.

Let $p_X$ be the probability mass function for $X$.

The **probability generating function for $X$**, denoted $\map {\Pi_X} s$, is the formal power series defined by:

- $\displaystyle \map {\Pi_X} s := \sum_{n \mathop = 0}^\infty \map {p_X} n s^n \in \R \left[\left[{s}\right]\right]$

### PID or pid

A **principal ideal domain** is an integral domain in which every ideal is a principal ideal.

### PMF or p.m.f.

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 $x$.

### PMI

### PNT

### Poset

A **partially ordered set** is a relational structure $\left({S, \preceq}\right)$ such that $\preceq$ is a partial ordering.

The **partially ordered set** $\left({S, \preceq}\right)$ is said to be **partially ordered by $\preceq$**.