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Area Problem01:26

Area Problem

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Determining the area of a region with straight edges is straightforward, as geometric formulas for rectangles, triangles, and polygons can be applied directly. However, traditional geometric methods are insufficient when a region has a curved boundary, such as the area under a function.fromThe area problem involves finding a systematic way to measure such regions. One approach to solving this problem is through approximation. Instead of attempting to compute the area exactly at the outset, the...
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Approximate Integration01:24

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In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
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Continuous probability distributions are used to model random variables that can take on any real value within a specified range. These variables do not take on isolated or countable values but rather exist on a continuum. For example, the height of an individual can be measured with increasing precision—such as 163.5 or 165.25 centimeters—demonstrating that height is a continuous random variable.The behavior of such variables is described using a probability density function (PDF),...
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Approximating areas under curved boundaries is a common problem in applied mathematics, particularly when an exact calculation is difficult or impractical. One effective numerical method for this purpose is the Midpoint Rule, which provides an estimate of the area under a curve by using rectangular approximations over a specified interval.Description of the Midpoint RuleThe Midpoint Rule begins by dividing the given interval into a number of equal subintervals. For each subinterval, the...
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 The probability of a random variable x  is the likelihood of its occurrence. A probability distribution represents the probabilities of a random variable using a formula, graph, or table. There are two types of probability distribution– discrete probability distribution and continuous probability distribution.
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Related Experiment Video

Updated: Jan 17, 2026

Single-Molecule Tracking Microscopy - A Tool for Determining the Diffusive States of Cytosolic Molecules
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Approximations of the cumulative distribution function using transport maps learning.

Dawen Wu1,2, Ludovic Chamoin3

  • 1CNRS@CREATE, 1 Create Way, #08-01 Create Tower, Singapore 138602, Singapore.

Chaos (Woodbury, N.Y.)
|September 19, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces Transport Map Learning (TML) for accurate, data-efficient approximations of cumulative distribution functions (CDFs). TML offers superior performance over traditional methods, especially when data is limited.

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Area of Science:

  • Statistics
  • Machine Learning
  • Numerical Analysis

Background:

  • Cumulative distribution functions (CDFs) for many probability distributions lack elementary closed-form expressions.
  • Existing approximation methods, like the empirical CDF, often require substantial sample data for accuracy.

Purpose of the Study:

  • To develop accurate and data-efficient closed-form approximations for CDFs.
  • To address the limitations of current CDF approximation techniques.

Main Methods:

  • Inspired by transport map theory, specifically the one-dimensional case where the transport map equals the CDF.
  • Proposed Transport Map Learning (TML) using a neural network trained to approximate the CDF.
  • Output of the neural network is passed through a sigmoid function to ensure a [0,1] range.

Main Results:

  • TML demonstrated high accuracy in approximating CDFs for standard normal, beta, and gamma distributions.
  • Achieved superior accuracy compared to empirical CDF methods with interpolation.
  • Effectiveness validated across benchmark probability distributions.

Conclusions:

  • Transport Map Learning (TML) provides a powerful, data-efficient approach for closed-form CDF approximation.
  • The method offers a significant improvement over existing techniques, particularly in data-scarce scenarios.
  • TML is a promising tool for statistical modeling and analysis where CDFs are critical.