Question about notation of expected value, $\mathbb{E}[1_{\{X^2+Y^2\leq{1}\}}]$

Question

I am self-teaching a course in Monte Carlo methods and have a quick question about notation. What is meant by this notation $\mathbb{E}[1_{\{X^2+Y^2\leq{1}\}}]$? I am familiar with expected value but have either forgotten or never encountered the notation $1_{\{X^2+Y^2\leq{1}\}}$. What does this mean? For context, here is the question I am attempting:

Let $X, Y$ be independent and uniformly distributed on $[0,1]$. Show that: $$\mathbb{E}[1_{\{X^2+Y^2\leq{1}\}}]=\frac{\pi}{4}.$$

Thanks in advance.

Answer 1

For any event $E$, the notation $1_E$ typically denotes the indicator function of $E$:

$$ 1_E = \begin{cases} 1 & E \text{ occurs}\\ 0 & \text{otherwise} \end{cases} $$

So, $$ 1_{X^2+Y^2 \leq 1} = \begin{cases} 1 & X^2+Y^2 \leq 1\\ 0 & \text{otherwise} \end{cases} $$