From ProofWiki
Jump to: navigation, search

Previous  ... Next


$\mathrm E$

The Système Internationale d'Unités metric scaling prefix denoting $10^{\, 18 }$.

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



$\mathrm E$ or $\mathrm e$

The hexadecimal digit $14$.

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




Used by some authors to denote a general set.

The $\LaTeX$ code for \(E\) is E .

Identity Element


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


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

$e$ is defined to be the unique real number such that the value of the (real) 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 .



$\mathcal E$

An experiment, which can conveniently be denoted $\mathcal E$, 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 .


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

Let $X$ be a discrete random variable.

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

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

whenever the sum is absolutely convergent, that is, when:

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

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:

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

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) .