Definition

Adverb 1. In a stochastic manner; by means of or according to stochastic processes: This describes an action done involving or characterized by random probability. It implies a method or result governed by the laws of chance rather than deterministic certainty.

Usage

The adverb "stochastically" is used to modify verbs, describing that an action is performed or a result is achieved through random processes. It is most common in academic, scientific, and technical contexts, particularly in mathematics, statistics, physics, and computer science.

Examples

• The model generates data stochastically to simulate real-world uncertainty.
• The particles' movements were determined stochastically, not by a fixed equation.
• We can stochastically sample from the population to estimate the overall trend.

Advanced Usage

• Stochastically Independent: A key concept in probability theory where the occurrence of one event does not affect the probability of another.
  • Example: In a fair model, each coin toss is stochastically independent of the last.
• Stochastically Dominant: Used in decision theory and economics to describe one random variable that is preferable to another under all possible outcomes for a rational decision-maker.
  • Example: Portfolio A is stochastically dominant over Portfolio B, meaning it offers a better chance of higher returns for any given risk level.

Variants and Related Words

• Stochastic (Adjective): Having a random probability distribution or pattern that may be analyzed statistically but not predicted precisely.
  • Example: The system is affected by stochastic noise.
• Stochasticity (Noun): The quality or state of being stochastic; randomness.
  • Example: The stochasticity of the process makes long-term prediction impossible.

Synonyms

• Randomly: In a way that lacks a definite pattern or purpose. (Note: "Randomly" is a broader, more general term, while "stochastically" often implies a specific, analyzable form of randomness governed by probability distributions.)
• Probabilistically: In a way that involves or is governed by probability theory.
• Indeterminately: In a way that is not fixed or known in advance.

Antonyms

• Deterministically: In a way where outcomes are precisely determined by initial conditions and fixed laws, with no randomness involved.
• Predictably: In a way that can be foreseen or expected with certainty.
• Systematically: According to a fixed plan or system; methodically.

Adverb

1. by stochastic means
   • we estimated the answer stochastically