This study reveals for the first time the essential mechanism of tool-integrated reasoning through rigorous mathematical proof and develops a novel training algorithm that enables AI to utilize tools ...

(The Conversation is an independent and nonprofit source of news, analysis and commentary from academic experts.) Toshi Hirabayashi, Georgia Institute of Technology (THE CONVERSATION) I was preparing ...

Efficient Sequential and Batch Estimation of Univariate and Bivariate Probability Density Functions and Cumulative Distribution Functions along with Quantiles (Univariate) and Nonparametric ...

A probability density function (PDF) describes the likelihood of different outcomes for a continuous random variable.

Probability is a measure of the likelihood of an event occurring, and it plays a significant role in many fields such as statistics, mathematics, and science. Understanding how to calculate ...

III. How to Calculate Continuous Probability Distribution? Calculating continuous probability distribution requires a more complex approach using integral calculus and different types of functions.

Given the characteristic function of a continuously distributed random variable, the authors showed that it is possible - using the Euler’s identity, to approximate pointwise the probability density ...