Free module · Quantitative toolkit

Statistics & Probability

Every price move is signal plus randomness. Statistics is how you tell them apart.

entry · validation

Lessons in this module

1. Start with counts, probabilities and averages
2. Random outcomes, sample averages and the limits of the bell curve
3. Conditional probability: update a belief with evidence
4. Prediction markets: probability, price and net expected value
5. Expected value: measure the payoff before choosing the risk
6. Correlation: how many strategies do you really have?
7. Conditional probability and Bayes: where the edge actually lives
8. The central limit theorem: sampling means under explicit assumptions
9. Law of large numbers: what averaging can and cannot establish
10. Central limit theorem, standard errors and confidence intervals
11. Total expectation and total variance: within-regime and between-regime risk
12. Markov, Chebyshev and Jensen: bounds without a full distribution
13. Gauss–Markov theorem: what a regression hedge ratio guarantees
14. Delta method: propagate estimator uncertainty through a smooth function
15. Multiple testing and the union bound: account for the whole search
16. Estimation uncertainty and Bayesian updating

Open the interactive module

Practice and apply

• From trade log to expectancy — 100 trades · 58 wins averaging +$180 · 42 losses averaging −$210
• Conditioning a win rate on a regime flag — P(win) = 0.55 · among winners, 65% occurred with VIX < 15 · among losers, 35% occurred with VIX < 15
• Law of large numbers: what averaging can and cannot establish — Evaluate (-2+1+3+2)/4 in dollars per trade.
• Central limit theorem, standard errors and confidence intervals — Evaluate 10/sqrt(100) in basis points per day of standard error.
• Total expectation and total variance: within-regime and between-regime risk — Evaluate 0.75*4+0.25*16+0.75*1+0.25*9 in squared basis points.
• Markov, Chebyshev and Jensen: bounds without a full distribution — Evaluate 1/2^2 in probability upper bound.
• Gauss–Markov theorem: what a regression hedge ratio guarantees — Evaluate 4/2 in dimensionless slope.
• Delta method: propagate estimator uncertainty through a smooth function — Evaluate 0.1/2^2 in inverse-rate units of standard error.
• Multiple testing and the union bound: account for the whole search — Evaluate 0.05/20 in p-value threshold.
• Count outcomes — A fair die has two outcomes greater than 4. Find their probability.
• Sampling error — 100 independent trades have sample SD $10. Estimate SE of the mean.
• Update from counts — Evidence appears in 80 stressed and 180 calm cases. What fraction of these cases are stressed?
• Contract expectancy — A $1 binary payout has probability .7, entry $.62 and total assumed cost $.01. Find expected profit.

Work through the practice exercises · Quant development tools