Symbols:E

Hexadecimal

$\mathrm E$ or $\mathrm e$

The hexadecimal digit $14$.

Its $\LaTeX$ code is \mathrm E or \mathrm e.

Set

$E$

Used by some authors to denote a general set.

The $\LaTeX$ code for $E$ is E.

Identity Element

$e$

Used to indicate the identity element in a general algebraic structure.

If $e$ is the identity of the structure $\left({S, \circ}\right)$, then a subscript is often used: $e_S$.

This is particularly common when more than one structure is under discussion.

The $\LaTeX$ code for $e_S$ is e_S.

Euler's number

$e$

Euler's number $e$ is the base of the natural logarithm $\ln$.

It is defined to be the unique real number such that the value of the exponential function $e^x$ has the same value as the slope of the tangent line to the graph of the function.

The $\LaTeX$ code for $e$ is e.

Experiment

$\mathcal E$

An experiment (or trial) is defined as:

a course of action whose consequence is not predetermined.

An experiment $\mathcal E$ can be formulated mathematically by means of a probability space, which consists of:

• The sample space $\Omega$: that is, the set of all possible outcomes of the experiment;

• The event space $\Sigma$: that is, the list of all the events which may occur as the consequences of the experiment;

• The probability measure $\Pr$ on the event space: that is, the likelihood of the happening of each of the events in the event space.

With this definition, $\mathcal E$ is a measure space $\left({\Omega, \Sigma, \Pr}\right)$ such that $\Pr \left({\Omega}\right) = 1$.

The $\LaTeX$ code for $\mathcal E$ is \mathcal E.

Expectation

$E \left({X}\right)$

Let $X$ be a discrete random variable.

The expectation of $X$ is written $E \left({X}\right)$, and is defined as:

$\displaystyle E \left({X}\right) := \sum_{x \in \operatorname{Im} \left({X}\right)} x \Pr \left({X = x}\right)$

whenever the sum is absolutely convergent, i.e. when:

$\displaystyle \sum_{x \in \operatorname{Im} \left({X}\right)} \left|{x \Pr \left({X = x}\right)}\right| < \infty$

The $\LaTeX$ code for $E \left({X}\right)$ is E \left({X}\right).

Conditional Expectation

$E \left({X \mid B}\right)$

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space.

Let $X$ be a discrete random variable on $\left({\Omega, \Sigma, \Pr}\right)$.

Let $B$ be an event in $\left({\Omega, \Sigma, \Pr}\right)$ such that $\Pr \left({B}\right) > 0$.

The conditional expectation of $X$ given $B$ is written $E \left({X \mid B}\right)$ and defined as:

$\displaystyle E \left({X \mid B}\right) = \sum_{x \in \operatorname{Im} \left({X}\right)} x \Pr \left({X = x \mid B}\right)$

whenever this sum converges absolutely.

Note that $\Pr \left({X = x \mid B}\right)$ denotes the conditional probability that $X = x$ given $B$.

The $\LaTeX$ code for $E \left({X \mid B}\right)$ is E \left({X \mid B}\right).