What is $\mathbb{1}$ with a subscript?

Question

In my textbook it's estimating $\mu = P(X > 2)$ with monte carlo estimation, and I'm confused about the line $\mu = P(X > 2) = E(\mathbb{1}_{\{X>2\}})$.

What would the $\mathbb{1}_{\{X>2\}}$ mean ?

It's used earlier as a remark "Let $A$ be a proper subset of $\mathbb{R}^d$. If $g(x) = \mathbb{1}_A(x)$, then $E(g(X)) = E(\mathbb{1}_A(X)) = 0 \times P(\mathbb{1}_A(x) = 0) + 1 \times P(\mathbb{1}_A(x) = 1) $

Answer 1

This is called an indicator function. Indicators return 1 if the condition is satisfied, zero otherwise.

So, when we write $1_{ \{X > 2\} }$, we mean that this function returns $1$ if the random variable $X$ has a value larger than $2$, and $0$ otherwise. Indicator functions can simplify some computations in probability.

Answer 2

and I'm confused about the line $\mu =\mathsf P(X>2)=\mathsf E(\mathbf 1_{{X>2}})$.

As noted in the comments and by Sean Roberson, you are dealing with an indicator function, or a characteristic function, which piecewise evaluates to $1$ when the indicated condition occurs but $0$ otherwise.   [The $1$ is typically emboldened; alternatively symbols such as $\Bbb I$ or $\raise{0.5ex}\chi$ may be used.]

$$\mathbf 1_{\{x>2\}}=\begin{cases}1&:&x>2\\0&:& \text{otherwise}\end{cases}$$

When the condition is an event, we often call this an indicator random variable.   The expected value of this is the probability that the event occurs; since by the definition of expectation :

$$\mathsf E(\mathbf 1_{\{X>2\}}) ~{= 1\cdot\mathsf P(X>2)+0\cdot \mathsf P(X\leq 2)\\=\mathsf P(X>2)}$$

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It's used earlier as a remark "Let A be a proper subset of Rd. If $g(x)=\mathbf 1_A(x)$, then $E(g(X))=E(\mathbf 1_A(X))=0×P(\mathbf 1_A(X)=0)+1×P(\mathbf 1_A(X)=1)$

This is an alternative notation that similarly notes when the argument is within the indicated domain. $$\mathbf 1_A(x)=\begin{cases} 1&:& x\in A\\0&:& x\notin A\end{cases}$$

Thus we have:

$$\mathsf E(\mathbf 1_A(X)) ~{= 1\cdot\mathsf P(\mathbf 1_A(X)=1)+0\cdot\mathsf P(\mathbf 1_A(X)=0)\\ = \mathsf P(X\in A)}$$

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I'd discourage using the two notations in the same text... but it happens.