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Related Concept Videos

Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
Applications of Integration to Probability Density Functions01:27

Applications of Integration to Probability Density Functions

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), which...
Propagation of Uncertainty from Random Error00:59

Propagation of Uncertainty from Random Error

An experiment often consists of more than a single step. In this case, measurements at each step give rise to uncertainty. Because the measurements occur in successive steps, the uncertainty in one step necessarily contributes to that in the subsequent step. As we perform statistical analysis on these types of experiments, we must learn to account for the propagation of uncertainty from one step to the next. The propagation of uncertainty depends on the type of arithmetic operation performed on...
The Entropy as a State Function01:14

The Entropy as a State Function

Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...
Probability Distributions01:32

Probability Distributions

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.
A discrete probability distribution is a probability distribution of discrete random variables. It can be categorized into binomial probability distribution and Poisson probability...
Reversible and Irreversible Processes01:14

Reversible and Irreversible Processes

The thermodynamic processes can be classified into reversible and irreversible processes. The processes that can be restored to their initial state are called reversible processes. It is only possible if the process is in quasi-static equilibrium, i.e., it takes place in infinitesimally small steps, and the system remains at equilibrium However, these are ideal processes and do not occur naturally. An ideal system undergoing a reversible process is always in thermodynamic equilibrium within...

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Related Experiment Video

Updated: Jun 19, 2026

Sealable Femtoliter Chamber Arrays for Cell-free Biology
13:44

Sealable Femtoliter Chamber Arrays for Cell-free Biology

Published on: March 11, 2015

A general definition of influence between stochastic processes.

Anne Gégout-Petit1, Daniel Commenges

  • 1IMB, UMR 5251, INRIA CQFD, Talence 33405, France.

Lifetime Data Analysis
|October 9, 2009
PubMed
Summary
This summary is machine-generated.

This study introduces a new definition for weak local conditional independence (WCLI) in stochastic processes, extending its applicability. It also defines strong local conditional independence (SCLI) for unambiguous causal influence analysis.

Related Experiment Videos

Last Updated: Jun 19, 2026

Sealable Femtoliter Chamber Arrays for Cell-free Biology
13:44

Sealable Femtoliter Chamber Arrays for Cell-free Biology

Published on: March 11, 2015

Area of Science:

  • * Stochastic processes
  • * Causal inference
  • * Statistical modeling

Background:

  • * The concept of weak local conditional independence (WCLI) was previously defined based on measurability conditions.
  • * Existing definitions may not cover a sufficiently broad range of stochastic processes for comprehensive causal analysis.

Purpose of the Study:

  • * To extend the definition of WCLI to a larger class of processes (D').
  • * To introduce a new definition of WCLI using likelihood processes and the Girsanov theorem.
  • * To define strong local conditional independence (SCLI) and explore its relationship with WCLI and conventional independence.

Main Methods:

  • * Extension of measurability conditions for WCLI.
  • * Application of the Girsanov theorem for likelihood-based WCLI definition.
  • * Construction of SCLI from WCLI.

Main Results:

  • * Two definitions of WCLI are presented and shown to coincide for the extended class of processes D' under specific conditions.
  • * WCLI and SCLI are defined, with SCLI being constructible from WCLI.
  • * Direct influence is indicated when WCLI fails, while SCLI failure implies direct or indirect influence.
  • * SCLI can be defined using conventional independence, but WCLI cannot.

Conclusions:

  • * The extended definitions of WCLI and SCLI provide a framework for analyzing causal influences in a broader class of stochastic processes.
  • * The distinction between WCLI and SCLI is crucial for differentiating direct versus direct/indirect influences.
  • * Mathematical definitions alone are insufficient for causal interpretation; a suitable system and true probability are necessary.