Why "Expected Value" Is Never What You Actually Get(The Math Behind Finance Ep6)

Every stock price you see is just one sample from a massive distribution — and that changes how options, risk, and AI all work.

In this episode of The Math Behind Finance, we build the probability foundations that power all of Phase 2: Bayes' theorem, expected value, variance and covariance, filtrations, and conditional expectation. You'll see a Gaussian distribution "breathe" as its σ changes, watch a regression curve emerge from 500 noisy data points, and discover why the exact same minimization — the argmin of mean squared error — powers both Black-Scholes option pricing AND the training of every neural network. By the end, you'll recognize the single master formula that governs every pricing model, hedging strategy, and supervised learning algorithm. This is the mathematical heart of everything quantitative finance and machine learning share.

━━━━━━━━━━━━━━━━━━━━━━━━━
CHAPTERS
━━━━━━━━━━━━━━━━━━━━━━━━━
0:00 Hook — One price, many futures
1:44 Bayes' Theorem & Conditional Probability
4:02 Expected Value & Linearity (no independence required!)
6:21 Variance, Covariance & the Breathing Gaussian
8:51 Filtration — Information Over Time
11:33 Conditional Expectation as the Best Predictor
13:55 One Formula, Two Worlds — Finance ↔ AI

━━━━━━━━━━━━━━━━━━━━━━━━━
SERIES & PLAYLISTS
━━━━━━━━━━━━━━━━━━━━━━━━━
📌 The Math Behind Finance:https://www.youtube.com/playlist?list=PLSjJDxqj7Cqx_Q81TYPi3vUjoR4Gt4Ysp
📌 The Math Behind AI (crossover series): https://www.youtube.com/playlist?list=PLSjJDxqj7Cqw9op4A2nGV5MxZi_Kf-qv3

━━━━━━━━━━━━━━━━━━━━━━━━━
ABOUT AXIOMMOTION
━━━━━━━━━━━━━━━━━━━━━━━━━
AxiomMotion makes visual, rigorous mathematics accessible to everyone. Each episode is built from first principles, every animation is independently verified, and every claim is checked against the primary literature. If that sounds like your kind of channel, subscribe for weekly math animations.

——
🛠️ Tools used in this video:
- Animation: Manim Community Edition (Python)
- Voice: ElevenLabs AI
- Manim Starter Pack (31 ready-to-use scenes): https://axiommotion.gumroad.com/l/drhyqd

#TheMathBehindFinance #ProbabilityTheory #QuantitativeFinance

Видео Why "Expected Value" Is Never What You Actually Get(The Math Behind Finance Ep6) канала AxiomMotion