Random walks in mathematics represent sequences of steps determined by chance, impacting physics, economics, and biology. They model unpredictable behaviors in systems like stock markets and particle motion. Key principles include the Markov property and the Central Limit Theorem, which aid in making predictions and understanding complex phenomena. The theory's real-world applications span finance, environmental modeling, and beyond, reflecting its versatility and importance.
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A mathematical concept that models the unpredictable behavior of systems using a sequence of steps determined by a random process
Physics
Helps explain the random motion of particles in a fluid
Economics
Used to model the unpredictable fluctuations of stock prices
Biology
Used to model the behavior of complex systems
Can be represented in various dimensions, from a simple line to multi-dimensional spaces
Assumes each step is equally likely to go in any direction
A metaphor for a random path taken by an inebriated person
Allow for modeling of more complex systems
Each step is independent and identically distributed, meaning the future state depends only on the present state
States that the sum of a large number of random variables will tend to follow a normal distribution
Suggests that steps or decisions within a system are random and follow no discernible pattern
Derived from random walk theory, it argues that asset prices cannot be consistently predicted from historical prices
Supports the use of probabilistic methods to forecast future events despite randomness
Used to model the unpredictable nature of stock price movements and incorporate market trends
Graphical representations show how individual trajectories can diverge significantly due to randomness
A more sophisticated random walk model used to represent natural and economic systems
Another advanced model used to better represent complex systems
Continues to adapt to new challenges and provide insights into stochastic aspects of the world
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Characteristics of random walk steps
Steps in a random walk are independent and identically distributed, with no step influenced by previous ones.
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Random walk in stock market application
Random walks model stock prices by reflecting unpredictable, non-patterned price changes over time.
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Brownian motion relation to random walks
Brownian motion exemplifies a random walk in physics, describing the random movement of particles in a fluid.
