Editing Stochastic Processes (section)

== <span style="color: #FFFFFF;">Understanding</span> ==
Stochastic processes are understood through '''Memory''' and '''Paths'''.

'''1. The "Memoryless" Property (Markov)''':
Does the past matter?
* In a **Markov Chain**, the answer is "No."
* If you are at a "Red Light," the chance of it turning "Green" is the same whether you arrived 1 second ago or 1 minute ago.
* This makes the math "Much simpler," allowing us to model everything from "Google PageRank" to "Sentence generation" in AI.

'''2. The Power of "Accumulation" (Random Walks)''':
A single random step is small, but a "Thousand steps" go far.
* In a 1D Random Walk, you are "Guaranteed" to eventually return to zero (your starting point).
* In a 3D Random Walk (like a "Bee" flying in the air), you are **NOT** guaranteed to ever find your way home.
* This explains why "Diffusing" chemicals spread out and never "Un-mix" themselves.

'''3. Continuous vs. Discrete Time''':
* **Discrete**: Things that change in "Steps" (e.g., the price of a stock at the end of every day).
* **Continuous**: Things that change "Constantly" (e.g., the temperature of a room).

'''The 'Google PageRank' Algorithm'''': The most famous application of Markov Chains. Google treats the "Whole Internet" as a giant State Space. A "Random Surfer" clicks links forever. The websites that the surfer "Ends up on" most often are the ones with the highest PageRank (and the highest "Stationary Distribution").
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