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I am currently trying to understand this paper. On page 4 in formula 2 I stumbled upon the letter: $\mathbb{E}$

As I understand it, this typeface is used to represent number sets. I have seen the more common ones (like $\mathbb{N}$) many times before, but have never encountered an E. What set of numbers is represented by the letter $\mathbb{E}$?

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  • $\begingroup$ The author is using $\mathbb{P}$ for "probability" and $\mathbb{E}$ for "expected value". In that formula, $I_{(*)}$ is the random variable that is $1$ if $(*)$ is true, otherwise $0$. The expected value of $I_{(*)}$ is the same thing as the probability of $(*)$. $\endgroup$ – Zach Teitler Oct 25 '17 at 11:56
  • $\begingroup$ Exactly, $I$ is a random variable given by the indicator function $I$. See en.wikipedia.org/wiki/Indicator_function. $\endgroup$ – Hector Blandin Oct 25 '17 at 12:02
  • $\begingroup$ Thanks for the clarifications! As this definition is not explicity mentioned in the paper, is it a standard? $\endgroup$ – Bananenaffe Oct 25 '17 at 12:14
  • $\begingroup$ Some notations are standard, like $\mathbb{N}$ for the natural numbers or $\mathbb{R}$ for the real numbers. But "standardness" of notation is actually rather rare. Just like $x$ and $f$ are used in many, many different ways in mathematics, so are most notations. So, what do you do about it? You read the book carefully and you look for where the author specifies the notations. What if the author doesn't specify the notations? I guess you ask us! $\endgroup$ – Lee Mosher Oct 25 '17 at 12:18
  • $\begingroup$ $E$ is pretty standard notation for expectation, but I've never seen $\mathbb{E}$ used for expectation. $\endgroup$ – Lee Mosher Oct 25 '17 at 12:18
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It is the expected value of a random variable.

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The notation $\mathbb{E}$ means the expectation of a random variable. Also $\mathbb{P}$ is a probability measure (see https://en.wikipedia.org/wiki/Probability_space). Ins this context $I$ is the indicator function. Let $\Omega$ be a probability space. Recall that if $X:\Omega\longrightarrow\mathbb{R}$ is a random variable it's expectation is defined as:

$$ \mathbb{E}(X):=\int_{\Omega}X(w)d\mathbb{P}(w),$$

where $\mathbb{P}$ is a probability measure (see https://en.wikipedia.org/wiki/Probability_measure). Also, if $A$ is a set, the notation $I$ means:

$$ I_{A}(t)=\begin{cases} 1 & \text{if}\ t\in A, \\ 0 & \text{otherwise.}\end{cases}$$

In particular, you will have:

$$\mathbb{E}(I_{A}(t))=\int_{\Omega}I_{A}(w)d\mathbb{P}(w)=\mathbb{P}(A).$$

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