Contract
- Input: problem description and inputs defined by the skill body.
- Output: Markdown artifact with completed process steps.
- Side effects: none.
- Dependencies: none.
- Stop condition: all process steps executed; artifact saved with required sections.
- Risk: low.
- Boundary: produces reasoning artifact only; no system changes.
Probability & Stochastic Modeling
Construct a probability model — distribution, process, or inference — with explicit assumptions and checks against known limits.
When to use
- User asks for random variables, distributions, stochastic processes, or inference.
- A risk / quant / ML problem needs a probabilistic backbone.
Process
1. Define the space
Name the sample space and events; identify if a sigma-algebra is needed (continuous spaces).
Completion criterion: sample space and event space explicit; independence / measurability called out if relevant.
2. Choose distribution / process
- Discrete (Bernoulli, Binomial, Poisson, Geometric).
- Continuous (Uniform, Normal, Exponential, Beta, Gamma, t, chi-square).
- Multivariate (Multivariate Normal, Dirichlet, Copulas).
- Stochastic processes (Markov chain, Poisson process, Brownian motion, Lévy, ARMA / GARCH).
- Bayesian: prior + likelihood → posterior.
Completion criterion: distribution or process named with type and parameters.
3. State parameters
Each parameter is named with value or estimation method (MLE, MAP, moment-matching, prior).
Completion criterion: parameters listed with values or estimation procedure.
4. Compute / simulate
- Analytical where tractable (mean, variance, CDF, moments).
- Monte Carlo with: seed, sample size N (justified), convergence check.
- For inference: posterior sampling (MCMC, variational, conjugate update).
Completion criterion: computation done; sample size or analytical result explicit.
5. Validate
- Limits: law of large numbers, central limit, stationary distribution, long-run mean.
- Sanity: small cases (n=1, 2, 3) checked by hand.
- Cross-check: analytical vs simulated mean / variance / tail probability.
- Calibration: does the model reproduce known data moments?
Completion criterion: validation completed; discrepancy addressed.
6. Deliver
Markdown artifact: model, parameters, computation, validation, and an operational interpretation ("expected return X% with std Y% over horizon T").
Completion criterion: artifact complete and reproducible.
1---2name: math-probability-models3description: Build probability models — distributions, stochastic processes (Markov, Brownian, Poisson), inference — with explicit assumptions and sanity checks.4---56## Contract78- **Input:** problem description and inputs defined by the skill body.9- **Output:** Markdown artifact with completed process steps.10- **Side effects:** none.11- **Dependencies:** none.12- **Stop condition:** all process steps executed; artifact saved with required sections.13- **Risk:** low.14- **Boundary:** produces reasoning artifact only; no system changes.151617# Probability & Stochastic Modeling1819Construct a **probability model** — distribution, process, or inference — with explicit assumptions and checks against known limits.2021## When to use2223- User asks for random variables, distributions, stochastic processes, or inference.24- A risk / quant / ML problem needs a probabilistic backbone.2526## Process2728### 1. Define the space2930Name the sample space and events; identify if a sigma-algebra is needed (continuous spaces).3132**Completion criterion:** sample space and event space explicit; independence / measurability called out if relevant.3334### 2. Choose distribution / process3536- Discrete (Bernoulli, Binomial, Poisson, Geometric).37- Continuous (Uniform, Normal, Exponential, Beta, Gamma, t, chi-square).38- Multivariate (Multivariate Normal, Dirichlet, Copulas).39- Stochastic processes (Markov chain, Poisson process, Brownian motion, Lévy, ARMA / GARCH).40- Bayesian: prior + likelihood → posterior.4142**Completion criterion:** distribution or process named with type and parameters.4344### 3. State parameters4546Each parameter is named with value or estimation method (MLE, MAP, moment-matching, prior).4748**Completion criterion:** parameters listed with values or estimation procedure.4950### 4. Compute / simulate5152- Analytical where tractable (mean, variance, CDF, moments).53- Monte Carlo with: seed, sample size N (justified), convergence check.54- For inference: posterior sampling (MCMC, variational, conjugate update).5556**Completion criterion:** computation done; sample size or analytical result explicit.5758### 5. Validate5960- **Limits:** law of large numbers, central limit, stationary distribution, long-run mean.61- **Sanity:** small cases (n=1, 2, 3) checked by hand.62- **Cross-check:** analytical vs simulated mean / variance / tail probability.63- **Calibration:** does the model reproduce known data moments?6465**Completion criterion:** validation completed; discrepancy addressed.6667### 6. Deliver6869Markdown artifact: model, parameters, computation, validation, and an operational interpretation ("expected return X% with std Y% over horizon T").7071**Completion criterion:** artifact complete and reproducible.
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