I was solving a question and in an intermediate step, I need to solve the following product:
$$\prod_{k=1}^N\mathbb{1}_{\{y_k\geq1\}}$$
Is there a way to simplify it or write it in a closed-form expression?
I could only think of the following expression:
$\mathbb{1}_A.\mathbb{1}_B = \mathbb{1}_{A\cap B}$, so the expression would look like $\mathbb{1}_{{\{y_1\geq1},{y_2\geq1},{y_3\geq1}\dots{y_N\geq1\}}}$
This question often arises in statistics for the likelihood with an uniform density.
Pending you know the value of each $y_i$, we can talk about order statistics : $y_{(1)}=\min_i y_i, \dots, y_{(N)}=\max_i y_i$.
Looking at your product, it is equal to 0 as soon as one $y_i$ is smaller than 1. Thus, you want the smallest $y_i$ to be bigger than 1. We have $$\displaystyle \prod_{k=1}^N \mathbb{1}_{\{y_k\ge1\}} = \mathbb{1}_{\{y_{(1)}\ge1\}}.$$
It is easier to maximize/minimize the likelihood now.