# How to obtain value range of empirical PDF, given the mode and area?

Given an empirical PDF of a continuous random variable $$X$$, then integrating over its entire defined domain will yield an area of size 1. To find the probability of $$X \ge x_1 \land X \le x_2$$ (as in the image), we can simply integrate the PDF over that range.

Since the area represents the cumulative probability of the range it is denoted by, we can ask questions like "Given a cumulative probability of 0.1 and the mode of the PDF, what is the range of the domain given an approach where we approximate $$PDF(x)$$ so that it demarcates the range for a known area?".

Question: How to choose $$y$$ of the blue line without having to do it numerically then?

Bonus question: How to handle multi-modal cases?

• The question can be asked, but answering it is a different ball game. You need to reconstruct a density function from very little information. As for multimodal cases, the "bonus" question underlines the difficulty. The fact that the same criteria could be satisfied by unimodal and multimodal cases shows that you don't have enough information to proceed. – Nick Cox Oct 23 '19 at 11:53