Stochastic Processes For Normal People

Analogy

Plot

Worked Example

Worked Example · equilibrium distribution

A website visitor is either browsing products or reading reviews. Each minute: 70% of browsers stay browsing, 30% switch to reviews. 60% of reviewers stay reading, 40% switch to browsing. After many minutes, what fraction are doing each?

work through the algebra →4

Standup Bit

Title

Memoryless

Premise

I trust Markov chains more than people because they admit they have no memory

Setup

The process only knows where it is right now — it has no idea how it got there

Punchline

Like being in a relationship where every morning is a clean slate

Which sounds romantic until you realize your partner forgot you were fighting and you have to re-explain why they're sleeping on the couch

hear the rest →

Misconception

intro textbooks

Common belief

A stochastic process is unpredictable because it involves randomness

Reality

Casinos stay profitable to the day, not the century

see what stochastic means →

Sandbox Code

python

Code

import numpy as np
import matplotlib.pyplot as plt

steps = np.random.choice([-1, 1], size=500)
walk = np.cumsum(steps)
plt.plot(walk)

run the walk — Five hundred steps of a particle that moves up or down by one unit each tick, fifty-fifty. The path wanders far from zero even though no direction is favoured — the surprise for most first-timers is how far it gets.

Key Point

The Key Point

A stochastic process is a collection of random variables indexed by time, where each variable's distribution can depend on earlier ones but the index set determines whether you use discrete or continuous methods

see the formal statement →

Concept

Concept · PROBABILITY

Random Walk

in the context of Stochastic Processes

A sequence of steps where each move direction is chosen purely by chance.

unpack it →

Concept

Concept · STATE MACHINES

Markov Property

in the context of Stochastic Processes

The rule where future behavior depends only on current state, ignoring past history.

unpack it →

Related Concepts

Random walkIto calculusPoisson jumpWhite noiseBrownian motion

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factStock prices follow geometric random paths

asidePageRank treats web browsing as random hops

questionHow do queues predict server crashes

factStock prices follow geometric random paths

asidePageRank treats web browsing as random hops

questionHow do queues predict server crashes

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Bullets

Outline

Anatomy of a stochastic model

▸State space definition
Discrete states
Continuous domains
▸Index set bounds
Discrete time steps
Continuous intervals
▸Probability space foundation

see the outline →

Book

Pushkin's Vowels

How Andrey Markov Scanned Twenty Thousand Letters of Verse to Map Chains of Pure Chance

Open the book

Key takeaways

How Andrey Markov Scanned Twenty Thousand Letters of Verse to Map Chains of Pure Chance

Lesson

Flipping Coins Across An Afternoon

subject: How randomness changes when time enters the picture.

START THE LESSON

Lesson

The Drunkard On The Street Corner

subject: Predicting the next step when memory resets every second.

START THE LESSON

Lesson

Dust Motes In A Sunbeam

subject: Why continuous jiggling needs its own mathematical machinery.

START THE LESSON

Lesson

Pricing Stock Options In Bad Weather

subject: Turning unpredictable drift into an exact contract price.

START THE LESSON

Lesson

Taming The Random Differential Equation

subject: How calculus survives when curves refuse to be smooth.

START THE LESSON

Cheat Sheet

Cheat Sheet

Transition Matrices At A Glance

The future depends only on where you stand right now.

The compact table that predicts long-term wandering states.

UNFOLD IT

Cheat Sheet

Cheat Sheet

Brownian Motion Quick Reference

Unpredictable variance scales with the square root of elapsed time.

Every variance rule and drift term on one card.

UNFOLD IT

Walkthrough

Simulating A Weather Chain In Code

Stepping through state probabilities one loop at a time.

WATCH IT DONE

Walkthrough

Tracing A Random Walk Across The Screen

Building a geometric Brownian path from bare random increments.

WATCH IT DONE

Practice

Sorting Pure Luck From Predictable Drift

Classify real scenarios before the clock runs out.

RUN THE DRILLS

Quiz

EXAMINATION BOOK

Quiz

Martingales And Stopping Times Checkup

Test your instinct on fair bets and exit thresholds.

TAKE THE TEST

Q&A

Q&AASK THE DESK

Common Traps In Real World Probability

Dear Desk, Why does my random simulation never match historical market volatility?

— A.

Straight answers to why stock charts baffle beginners.

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