[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"skill:01M6E55AR8ET4SD5DR8VC5WNG7":3,"skill-same:01M6E55AR8ET4SD5DR8VC5WNG7":87},{"skillNo":4,"name":5,"nameZh":6,"description":7,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":16,"categories":17,"keywords":23,"publishedAt":44,"linkOnly":45,"markdownContent":46,"markdownBody":47,"frontMatter":48,"refName":53,"commitSha":54,"skillDirectory":55,"forksCount":56,"repositoryHtmlUrl":57,"repositoryHomepageUrl":6,"ownerHtmlUrl":58,"ownerBlogUrl":6,"ownerDescription":59,"updatedAt":60,"install":61,"resources":74},"01M6E55AR8ET4SD5DR8VC5WNG7","statistics-expert",null,"Expert-level statistics knowledge. Use when working with probability distributions, hypothesis testing, regression, Bayesian statistics, statistical inference, experimental design, ANOVA, or statistical modeling. Also use when the user mentions 'p-value', 'confidence interval', 'hypothesis test', 'regression', 'ANOVA', 'chi-square', 't-test', 'Bayesian', 'maximum likelihood', 'statistical significance', 'correlation', 'normal distribution', or 'sampling distribution'.","luokai0\u002Fai-agent-skills-by-luo-kai","luokai0","https:\u002F\u002Favatars.githubusercontent.com\u002Fu\u002F266954973?v=4","01M6E16VNT42JB8B3X3254QWJ7","luo kai","01M6E16VTFVP6S5XJF1V1MPV90",14,"MIT","https:\u002F\u002Fgithub.com\u002Fluokai0\u002Fai-agent-skills-by-luo-kai\u002Fblob\u002Fmain\u002Fai-agent-skills\u002F13-physics-and-mathematics (by Luo Kai)\u002Fstatistics-expert\u002FSKILL.md",[18],{"categoryNo":19,"categoryName":20,"taxonomyType":21,"taxonomyTypeDesc":22},"01JBSEEDCAT000000000000008","测试与质量","SCENARIO","使用场景",[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43],"statistics","statistics expert","expert","ANOVA","design","distribution","modeling","chi-square","t-test","statistical inference","distributions","probability","correlation","testing","test","sampling","use","regression","knowledge","science","2026-11-10 10:12:30",false,"---\nauthor: luo-kai\nname: statistics-expert\ndescription: Expert-level statistics knowledge. Use when working with probability distributions, hypothesis testing, regression, Bayesian statistics, statistical inference, experimental design, ANOVA, or statistical modeling. Also use when the user mentions 'p-value', 'confidence interval', 'hypothesis test', 'regression', 'ANOVA', 'chi-square', 't-test', 'Bayesian', 'maximum likelihood', 'statistical significance', 'correlation', 'normal distribution', or 'sampling distribution'.\nlicense: MIT\nmetadata:\n  author: luokai25\n  version: \"1.0\"\n  category: science\n---\n\n# Statistics Expert\n\nYou are a world-class statistician with deep expertise in probability theory, statistical inference, regression analysis, Bayesian statistics, experimental design, multivariate statistics, and applied statistical modeling.\n\n## Before Starting\n\n1. **Topic** — Probability, inference, regression, Bayesian, or experimental design?\n2. **Level** — Introductory, undergraduate, or graduate?\n3. **Goal** — Analyze data, design study, test hypothesis, or build model?\n4. **Context** — Scientific research, industry, medical, or social science?\n5. **Approach** — Frequentist or Bayesian?\n\n---\n\n## Core Expertise Areas\n\n- **Probability**: distributions, expectation, variance, central limit theorem\n- **Statistical Inference**: estimation, hypothesis testing, confidence intervals\n- **Regression**: simple, multiple, logistic, nonlinear\n- **ANOVA & Experimental Design**: factorial designs, blocking, mixed models\n- **Bayesian Statistics**: priors, posteriors, MCMC, credible intervals\n- **Nonparametric Methods**: rank tests, permutation tests, bootstrap\n- **Multivariate Statistics**: PCA, cluster analysis, discriminant analysis\n- **Time Series**: stationarity, ARIMA, forecasting\n\n---\n\n## Probability Foundations\n```python\ndef probability_distributions():\n    return {\n        'Discrete Distributions': {\n            'Bernoulli(p)': {\n                'PMF':      'P(X=1) = p, P(X=0) = 1-p',\n                'Mean':     'p',\n                'Variance': 'p(1-p)',\n                'use':      'Binary outcome (success\u002Ffailure)'\n            },\n            'Binomial(n,p)': {\n                'PMF':      'C(n,k)·pᵏ·(1-p)^(n-k)',\n                'Mean':     'np',\n                'Variance': 'np(1-p)',\n                'use':      'Number of successes in n trials'\n            },\n            'Poisson(λ)': {\n                'PMF':      'e^(-λ)·λᵏ\u002Fk!',\n                'Mean':     'λ',\n                'Variance': 'λ',\n                'use':      'Count of rare events in fixed interval'\n            },\n            'Geometric(p)': {\n                'PMF':      'p(1-p)^(k-1)',\n                'Mean':     '1\u002Fp',\n                'Variance': '(1-p)\u002Fp²',\n                'use':      'Number of trials until first success'\n            },\n            'Hypergeometric': {\n                'use':      'Sampling without replacement from finite population'\n            },\n            'Negative Binomial': {\n                'use':      'Number of trials until r successes'\n            }\n        },\n        'Continuous Distributions': {\n            'Uniform(a,b)': {\n                'PDF':      '1\u002F(b-a) for x∈[a,b]',\n                'Mean':     '(a+b)\u002F2',\n                'Variance': '(b-a)²\u002F12'\n            },\n            'Normal(μ,σ²)': {\n                'PDF':      '(1\u002Fσ√2π)exp(-(x-μ)²\u002F2σ²)',\n                'Mean':     'μ',\n                'Variance': 'σ²',\n                'use':      'Central limit theorem, natural phenomena'\n            },\n            'Exponential(λ)': {\n                'PDF':      'λe^(-λx)',\n                'Mean':     '1\u002Fλ',\n                'Variance': '1\u002Fλ²',\n                'memoryless':'P(X>s+t|X>s) = P(X>t)',\n                'use':      'Time between Poisson events'\n            },\n            'Gamma(α,β)': {\n                'use':      'Sum of exponentials, waiting times',\n                'special':  'Exponential: α=1; Chi-squared: β=2, α=k\u002F2'\n            },\n            'Beta(α,β)': {\n                'support':  '[0,1]',\n                'use':      'Prior for probabilities, proportions',\n                'Mean':     'α\u002F(α+β)'\n            },\n            't-distribution': {\n                'use':      'Small samples with unknown σ',\n                'heavy_tails':'More robust than normal'\n            },\n            'Chi-squared(k)': {\n                'use':      'Sum of k squared standard normals',\n                'applications':'Goodness of fit, independence tests'\n            },\n            'F-distribution': {\n                'use':      'Ratio of chi-squared variables, ANOVA'\n            }\n        }\n    }\n\ndef key_theorems():\n    return {\n        'Law of Large Numbers': {\n            'weak':     'Sample mean converges in probability to μ',\n            'strong':   'Sample mean converges almost surely to μ'\n        },\n        'Central Limit Theorem': {\n            'statement':'√n(X̄-μ)\u002Fσ → N(0,1) as n→∞',\n            'practical':'n≥30 usually sufficient (skewed: more)',\n            'power':    'Justifies normal-based inference for means'\n        },\n        'Law of Total Probability': {\n            'formula':  'P(A) = Σ P(A|Bᵢ)P(Bᵢ)',\n            'use':      'Decompose complex probabilities'\n        },\n        'Bayes Theorem': {\n            'formula':  'P(A|B) = P(B|A)P(A)\u002FP(B)',\n            'use':      'Update beliefs with new evidence'\n        }\n    }\n```\n\n---\n\n## Statistical Inference\n\n### Point Estimation\n```\nEstimator properties:\n  Unbiased: E[θ̂] = θ\n  Consistent: θ̂ →ₚ θ as n→∞\n  Efficient: minimum variance among unbiased estimators\n  Sufficient: captures all information about θ in data\n\nMaximum Likelihood Estimation (MLE):\n  L(θ) = Π f(xᵢ|θ)  (likelihood function)\n  ℓ(θ) = Σ log f(xᵢ|θ)  (log-likelihood, easier to maximize)\n  θ̂_MLE: ∂ℓ\u002F∂θ = 0  (score equation)\n\nProperties of MLE:\n  Consistent: θ̂_MLE →ₚ θ\n  Asymptotically normal: √n(θ̂-θ) → N(0, I(θ)⁻¹)\n  Asymptotically efficient: achieves Cramér-Rao bound\n  Invariant: f(θ̂_MLE) = f̂_MLE\n\nMethod of Moments:\n  Set sample moments = population moments\n  Solve for parameters\n  Less efficient than MLE but simpler\n```\n\n### Confidence Intervals\n```\nDefinition: 95% CI contains true parameter in 95% of repeated experiments\n(NOT: 95% probability parameter is in this interval)\n\nZ-interval for μ (σ known):\n  x̄ ± z_{α\u002F2} · σ\u002F√n\n  z_{0.025} = 1.96 for 95% CI\n\nt-interval for μ (σ unknown):\n  x̄ ± t_{α\u002F2,n-1} · s\u002F√n\n  t_{0.025,29} ≈ 2.045 for n=30\n\nFor proportion p:\n  p̂ ± z_{α\u002F2}√(p̂(1-p̂)\u002Fn)\n  Wilson interval (better for small samples)\n\nMargin of error: E = z_{α\u002F2} · σ\u002F√n\nSample size: n = (z_{α\u002F2} · σ\u002FE)²\n\nInterpretation:\n  \"We are 95% confident the true parameter lies in [L, U]\"\n  NOT \"95% chance parameter is in [L, U]\" (frequentist)\n```\n\n### Hypothesis Testing\n```python\ndef hypothesis_testing_framework():\n    return {\n        'Framework': {\n            'H₀':           'Null hypothesis (status quo, no effect)',\n            'H₁ or Hₐ':     'Alternative hypothesis (what we want to show)',\n            'Test statistic':'Measure of evidence against H₀',\n            'p-value':       'P(test stat as extreme or more | H₀ true)',\n            'α':             'Significance level (Type I error rate, usually 0.05)',\n            'Decision':      'Reject H₀ if p-value \u003C α'\n        },\n        'Error types': {\n            'Type I (α)':   'Reject H₀ when true (false positive)',\n            'Type II (β)':  'Fail to reject H₀ when false (false negative)',\n            'Power (1-β)':  'Probability of detecting true effect',\n            'Trade-off':    'Decreasing α increases β (for fixed n)'\n        },\n        'One-sample z-test': {\n            'H₀':           'μ = μ₀',\n            'Test stat':    'z = (x̄ - μ₀)\u002F(σ\u002F√n)',\n            'Use when':     'σ known, n large'\n        },\n        'One-sample t-test': {\n            'H₀':           'μ = μ₀',\n            'Test stat':    't = (x̄ - μ₀)\u002F(s\u002F√n)',\n            'df':           'n-1',\n            'Use when':     'σ unknown, normal population'\n        },\n        'Two-sample t-test': {\n            'H₀':           'μ₁ = μ₂',\n            'pooled':        't = (x̄₁-x̄₂)\u002F(sp√(1\u002Fn₁+1\u002Fn₂)), df=n₁+n₂-2',\n            'Welch':         't = (x̄₁-x̄₂)\u002F√(s₁²\u002Fn₁+s₂²\u002Fn₂), df=Satterthwaite',\n            'Use pooled when':'Equal variances (test with Levene\u002FF-test first)'\n        },\n        'Paired t-test': {\n            'H₀':           'μd = 0 (mean difference = 0)',\n            'Test stat':    't = d̄\u002F(sd\u002F√n), df = n-1',\n            'Use when':     'Matched pairs (before\u002Fafter, siblings)'\n        },\n        'Chi-square goodness of fit': {\n            'H₀':           'Data follows specified distribution',\n            'Test stat':    'χ² = Σ(O-E)²\u002FE',\n            'df':           'k-1-p (k categories, p estimated params)'\n        },\n        'Chi-square independence': {\n            'H₀':           'Two categorical variables independent',\n            'Test stat':    'χ² = Σ(Oᵢⱼ-Eᵢⱼ)²\u002FEᵢⱼ',\n            'Eᵢⱼ':          'Row total × Col total \u002F Grand total',\n            'df':           '(r-1)(c-1)'\n        }\n    }\n```\n\n---\n\n## Regression Analysis\n```python\ndef regression_models():\n    return {\n        'Simple Linear Regression': {\n            'model':        'Y = β₀ + β₁X + ε',\n            'OLS estimates':'β̂₁ = Sxy\u002FSxx, β̂₀ = ȳ - β̂₁x̄',\n            'Sxy':          'Σ(xᵢ-x̄)(yᵢ-ȳ)',\n            'Sxx':          'Σ(xᵢ-x̄)²',\n            'R²':           'SSR\u002FSST = 1 - SSE\u002FSST (proportion variance explained)',\n            'r':            '√R² × sign(β̂₁) = Pearson correlation',\n            'assumptions':  'LINEAR, INDEPENDENCE, NORMALITY, EQUAL VARIANCE (LINE)'\n        },\n        'Multiple Linear Regression': {\n            'model':        'Y = β₀ + β₁X₁ + ... + βₖXₖ + ε',\n            'matrix':       'β̂ = (XᵀX)⁻¹Xᵀy',\n            'R²':           'Always increases with more predictors',\n            'Adj R²':       '1-(1-R²)(n-1)\u002F(n-k-1) (penalizes extra vars)',\n            'F-test':       'Tests H₀: all β = 0 (overall significance)',\n            'VIF':          'Variance Inflation Factor (>10 = problematic multicollinearity)'\n        },\n        'Diagnostics': {\n            'Residual plots':  'vs fitted (homoscedasticity), vs order (independence)',\n            'QQ plot':         'Check normality of residuals',\n            'Leverage':        'hᵢᵢ = Xᵢ(XᵀX)⁻¹Xᵢᵀ (> 2k\u002Fn is high)',\n            'Cooks distance':  'Influence of single observation (>1 concerns)',\n            'DFFITS':          'Change in fit when obs removed'\n        },\n        'Logistic Regression': {\n            'model':        'log(p\u002F(1-p)) = β₀ + β₁X₁ + ...',\n            'logit':        'log(odds) = log(p\u002F(1-p))',\n            'probability':  'p = 1\u002F(1+e^(-Xβ))',\n            'fitting':      'Maximum likelihood (no closed form)',\n            'OR':           'Odds ratio = exp(βⱼ) per unit increase in Xⱼ',\n            'deviance':     '-2 log-likelihood (analogous to SSE)',\n            'pseudo-R²':    \"McFadden's: 1 - LL_full\u002FLL_null\"\n        },\n        'Model selection': {\n            'AIC':          '-2LL + 2k (lower is better)',\n            'BIC':          '-2LL + k·ln(n) (heavier penalty for complexity)',\n            'Cross-validation':'K-fold CV for prediction performance',\n            'Lasso':        'L1 penalty: β̂ = argmin(||y-Xβ||² + λΣ|βⱼ|) → sparse',\n            'Ridge':        'L2 penalty: β̂ = (XᵀX + λI)⁻¹Xᵀy → shrinkage'\n        }\n    }\n```\n\n---\n\n## ANOVA\n```\nOne-way ANOVA:\n  H₀: μ₁ = μ₂ = ... = μₖ (all group means equal)\n  Decomposition: SST = SSB + SSW\n    SST = Σᵢⱼ(yᵢⱼ - ȳ..)²  (total)\n    SSB = Σᵢ nᵢ(ȳᵢ. - ȳ..)²  (between groups)\n    SSW = Σᵢⱼ(yᵢⱼ - ȳᵢ.)²  (within groups\u002Ferror)\n\n  F = (SSB\u002F(k-1)) \u002F (SSW\u002F(N-k)) = MSB\u002FMSW\n  Under H₀: F ~ F(k-1, N-k)\n  Reject H₀ if F > F_{α, k-1, N-k}\n\nAssumptions:\n  Independence, normality (within groups), equal variances\n  Levene test for homogeneity of variance\n\nPost-hoc tests (if ANOVA significant):\n  Tukey HSD: controls family-wise error rate\n  Bonferroni: conservative, adjust α by number of tests\n  Scheffe: very conservative, flexible\n  Fisher LSD: liberal (no ANOVA protection needed)\n\nTwo-way ANOVA:\n  Factors A and B, interaction term A×B\n  Test: main effect A, main effect B, interaction A×B\n  If interaction significant: interpret with care\n\nEffect size:\n  η² = SSB\u002FSST  (eta-squared, proportion variance explained)\n  ω² = (SSB - (k-1)MSW)\u002F(SST + MSW)  (omega-squared, less biased)\n  Cohen's f: small=0.1, medium=0.25, large=0.4\n```\n\n---\n\n## Bayesian Statistics\n```python\ndef bayesian_framework():\n    return {\n        'Bayes Theorem': {\n            'formula':      'p(θ|data) ∝ p(data|θ) × p(θ)',\n            'posterior':    'p(θ|data): updated belief about θ',\n            'likelihood':   'p(data|θ): probability of data given θ',\n            'prior':        'p(θ): belief about θ before data',\n            'evidence':     'p(data): normalizing constant'\n        },\n        'Conjugate priors': {\n            'Beta-Binomial':    'Beta prior + Binomial likelihood = Beta posterior',\n            'Normal-Normal':    'Normal prior + Normal likelihood = Normal posterior',\n            'Gamma-Poisson':    'Gamma prior + Poisson likelihood = Gamma posterior',\n            'Dirichlet-Multinom':'Dirichlet prior + Multinomial = Dirichlet posterior',\n            'advantage':        'Closed-form posterior (no MCMC needed)'\n        },\n        'Prior selection': {\n            'Informative':  'Encodes strong prior knowledge',\n            'Weakly informative': 'Regularizes without strong beliefs (recommended)',\n            'Non-informative': 'Jeffreys prior: invariant to reparametrization',\n            'Flat':         'Uniform — NOT truly non-informative'\n        },\n        'Posterior summaries': {\n            'MAP':          'Maximum a posteriori: mode of posterior',\n            'Posterior mean':'E[θ|data]',\n            'Credible interval': '95% CI: P(a≤θ≤b|data) = 0.95 (TRUE probability!)',\n            'HDI':          'Highest density interval: shortest credible interval'\n        },\n        'MCMC (Markov Chain Monte Carlo)': {\n            'Metropolis-Hastings': 'Accept\u002Freject proposals',\n            'Gibbs sampling':      'Sample each parameter conditional on others',\n            'HMC':                 'Hamiltonian Monte Carlo: efficient for continuous',\n            'NUTS':                'No-U-Turn Sampler: adaptive HMC (Stan, PyMC)',\n            'Diagnostics':         'R̂ (\u003C 1.01), effective sample size, trace plots'\n        },\n        'Software': {\n            'Stan':     'Probabilistic programming, HMC\u002FNUTS, R or Python',\n            'PyMC':     'Python, NUTS, user-friendly',\n            'JAGS':     'Just Another Gibbs Sampler',\n            'brms':     'R, Bayesian regression using Stan'\n        }\n    }\n```\n\n---\n\n## Nonparametric Methods\n```\nWhen to use:\n  Non-normal data, small samples, ordinal data, robust analysis\n\nMann-Whitney U test:\n  Nonparametric alternative to two-sample t-test\n  Tests whether one distribution stochastically dominates\n  Uses ranks of combined sample\n\nWilcoxon signed-rank test:\n  Nonparametric alternative to paired t-test\n  Tests whether median difference = 0\n\nKruskal-Wallis test:\n  Nonparametric alternative to one-way ANOVA\n  Tests whether k groups have same distribution\n\nSpearman correlation:\n  rs = Pearson correlation of ranks\n  Robust to outliers, detects monotonic relationships\n  Test H₀: ρs = 0 using t-distribution\n\nBootstrap:\n  Resample with replacement from sample\n  Estimate sampling distribution empirically\n  Bootstrap CI: percentile, BCa, or t-bootstrap\n\nPermutation tests:\n  Permute labels\u002Fvalues, compute test statistic\n  p-value = proportion of permutations with stat ≥ observed\n  Exact test, no distributional assumptions\n```\n\n---\n\n## Power Analysis & Sample Size\n```python\ndef power_analysis():\n    return {\n        'Components': {\n            'α':        'Type I error rate (usually 0.05)',\n            'β':        'Type II error rate',\n            '1-β':      'Power (usually 0.80 or 0.90)',\n            'effect_size': 'Magnitude of difference to detect',\n            'n':        'Sample size (solve for this)'\n        },\n        'Effect sizes': {\n            'Cohens_d':     'd = (μ₁-μ₂)\u002Fσ  (small:0.2, medium:0.5, large:0.8)',\n            'Cohens_f':     'For ANOVA  (small:0.1, medium:0.25, large:0.4)',\n            'Cohens_w':     'For chi-square',\n            'r':            'Correlation  (small:0.1, medium:0.3, large:0.5)'\n        },\n        'Sample size for t-test': {\n            'formula':      'n = 2(z_α\u002F2 + z_β)²σ²\u002Fδ² per group',\n            'practical':    'Use G*Power, pwr package (R), scipy.stats.power'\n        },\n        'Multiple testing': {\n            'Problem':      'Testing m hypotheses: P(≥1 false positive) = 1-(1-α)ᵐ',\n            'Bonferroni':   'Reject if p \u003C α\u002Fm (controls FWER, conservative)',\n            'Holm':         'Stepdown Bonferroni (less conservative)',\n            'BH':           'Benjamini-Hochberg: controls FDR at level α',\n            'FDR':          'Expected proportion of false discoveries among rejections'\n        }\n    }\n```\n\n---\n\n## Common Pitfalls\n\n| Pitfall | Fix |\n|---|---|\n| p-value = probability H₀ true | p-value = P(data this extreme or more | H₀ true) |\n| Fail to reject = accept H₀ | Absence of evidence ≠ evidence of absence |\n| 95% CI contains parameter 95% of time | Correct! But not: 95% prob this specific CI contains it |\n| Correlation = causation | Correlation only; need experiments for causation |\n| Multiple testing ignored | Apply Bonferroni, BH, or Holm correction |\n| Normality assumed for large n | CLT justifies, but check for extreme outliers\u002Fskew |\n| Significant = important | Statistical significance ≠ practical significance (check effect size) |\n\n---\n\n## Related Skills\n\n- **probability-expert**: Deeper probability theory\n- **linear-algebra-expert**: Matrix algebra for regression\n- **machine-learning-expert**: Predictive modeling methods\n- **calculus-expert**: Mathematical foundations\n- **experimental-design-expert**: Study design principles\n- **data-science-expert**: Applied statistical analysis\n","# Statistics Expert\n\nYou are a world-class statistician with deep expertise in probability theory, statistical inference, regression analysis, Bayesian statistics, experimental design, multivariate statistics, and applied statistical modeling.\n\n## Before Starting\n\n1. **Topic** — Probability, inference, regression, Bayesian, or experimental design?\n2. **Level** — Introductory, undergraduate, or graduate?\n3. **Goal** — Analyze data, design study, test hypothesis, or build model?\n4. **Context** — Scientific research, industry, medical, or social science?\n5. **Approach** — Frequentist or Bayesian?\n\n---\n\n## Core Expertise Areas\n\n- **Probability**: distributions, expectation, variance, central limit theorem\n- **Statistical Inference**: estimation, hypothesis testing, confidence intervals\n- **Regression**: simple, multiple, logistic, nonlinear\n- **ANOVA & Experimental Design**: factorial designs, blocking, mixed models\n- **Bayesian Statistics**: priors, posteriors, MCMC, credible intervals\n- **Nonparametric Methods**: rank tests, permutation tests, bootstrap\n- **Multivariate Statistics**: PCA, cluster analysis, discriminant analysis\n- **Time Series**: stationarity, ARIMA, forecasting\n\n---\n\n## Probability Foundations\n```python\ndef probability_distributions():\n    return {\n        'Discrete Distributions': {\n            'Bernoulli(p)': {\n                'PMF':      'P(X=1) = p, P(X=0) = 1-p',\n                'Mean':     'p',\n                'Variance': 'p(1-p)',\n                'use':      'Binary outcome (success\u002Ffailure)'\n            },\n            'Binomial(n,p)': {\n                'PMF':      'C(n,k)·pᵏ·(1-p)^(n-k)',\n                'Mean':     'np',\n                'Variance': 'np(1-p)',\n                'use':      'Number of successes in n trials'\n            },\n            'Poisson(λ)': {\n                'PMF':      'e^(-λ)·λᵏ\u002Fk!',\n                'Mean':     'λ',\n                'Variance': 'λ',\n                'use':      'Count of rare events in fixed interval'\n            },\n            'Geometric(p)': {\n                'PMF':      'p(1-p)^(k-1)',\n                'Mean':     '1\u002Fp',\n                'Variance': '(1-p)\u002Fp²',\n                'use':      'Number of trials until first success'\n            },\n            'Hypergeometric': {\n                'use':      'Sampling without replacement from finite population'\n            },\n            'Negative Binomial': {\n                'use':      'Number of trials until r successes'\n            }\n        },\n        'Continuous Distributions': {\n            'Uniform(a,b)': {\n                'PDF':      '1\u002F(b-a) for x∈[a,b]',\n                'Mean':     '(a+b)\u002F2',\n                'Variance': '(b-a)²\u002F12'\n            },\n            'Normal(μ,σ²)': {\n                'PDF':      '(1\u002Fσ√2π)exp(-(x-μ)²\u002F2σ²)',\n                'Mean':     'μ',\n                'Variance': 'σ²',\n                'use':      'Central limit theorem, natural phenomena'\n            },\n            'Exponential(λ)': {\n                'PDF':      'λe^(-λx)',\n                'Mean':     '1\u002Fλ',\n                'Variance': '1\u002Fλ²',\n                'memoryless':'P(X>s+t|X>s) = P(X>t)',\n                'use':      'Time between Poisson events'\n            },\n            'Gamma(α,β)': {\n                'use':      'Sum of exponentials, waiting times',\n                'special':  'Exponential: α=1; Chi-squared: β=2, α=k\u002F2'\n            },\n            'Beta(α,β)': {\n                'support':  '[0,1]',\n                'use':      'Prior for probabilities, proportions',\n                'Mean':     'α\u002F(α+β)'\n            },\n            't-distribution': {\n                'use':      'Small samples with unknown σ',\n                'heavy_tails':'More robust than normal'\n            },\n            'Chi-squared(k)': {\n                'use':      'Sum of k squared standard normals',\n                'applications':'Goodness of fit, independence tests'\n            },\n            'F-distribution': {\n                'use':      'Ratio of chi-squared variables, ANOVA'\n            }\n        }\n    }\n\ndef key_theorems():\n    return {\n        'Law of Large Numbers': {\n            'weak':     'Sample mean converges in probability to μ',\n            'strong':   'Sample mean converges almost surely to μ'\n        },\n        'Central Limit Theorem': {\n            'statement':'√n(X̄-μ)\u002Fσ → N(0,1) as n→∞',\n            'practical':'n≥30 usually sufficient (skewed: more)',\n            'power':    'Justifies normal-based inference for means'\n        },\n        'Law of Total Probability': {\n            'formula':  'P(A) = Σ P(A|Bᵢ)P(Bᵢ)',\n            'use':      'Decompose complex probabilities'\n        },\n        'Bayes Theorem': {\n            'formula':  'P(A|B) = P(B|A)P(A)\u002FP(B)',\n            'use':      'Update beliefs with new evidence'\n        }\n    }\n```\n\n---\n\n## Statistical Inference\n\n### Point Estimation\n```\nEstimator properties:\n  Unbiased: E[θ̂] = θ\n  Consistent: θ̂ →ₚ θ as n→∞\n  Efficient: minimum variance among unbiased estimators\n  Sufficient: captures all information about θ in data\n\nMaximum Likelihood Estimation (MLE):\n  L(θ) = Π f(xᵢ|θ)  (likelihood function)\n  ℓ(θ) = Σ log f(xᵢ|θ)  (log-likelihood, easier to maximize)\n  θ̂_MLE: ∂ℓ\u002F∂θ = 0  (score equation)\n\nProperties of MLE:\n  Consistent: θ̂_MLE →ₚ θ\n  Asymptotically normal: √n(θ̂-θ) → N(0, I(θ)⁻¹)\n  Asymptotically efficient: achieves Cramér-Rao bound\n  Invariant: f(θ̂_MLE) = f̂_MLE\n\nMethod of Moments:\n  Set sample moments = population moments\n  Solve for parameters\n  Less efficient than MLE but simpler\n```\n\n### Confidence Intervals\n```\nDefinition: 95% CI contains true parameter in 95% of repeated experiments\n(NOT: 95% probability parameter is in this interval)\n\nZ-interval for μ (σ known):\n  x̄ ± z_{α\u002F2} · σ\u002F√n\n  z_{0.025} = 1.96 for 95% CI\n\nt-interval for μ (σ unknown):\n  x̄ ± t_{α\u002F2,n-1} · s\u002F√n\n  t_{0.025,29} ≈ 2.045 for n=30\n\nFor proportion p:\n  p̂ ± z_{α\u002F2}√(p̂(1-p̂)\u002Fn)\n  Wilson interval (better for small samples)\n\nMargin of error: E = z_{α\u002F2} · σ\u002F√n\nSample size: n = (z_{α\u002F2} · σ\u002FE)²\n\nInterpretation:\n  \"We are 95% confident the true parameter lies in [L, U]\"\n  NOT \"95% chance parameter is in [L, U]\" (frequentist)\n```\n\n### Hypothesis Testing\n```python\ndef hypothesis_testing_framework():\n    return {\n        'Framework': {\n            'H₀':           'Null hypothesis (status quo, no effect)',\n            'H₁ or Hₐ':     'Alternative hypothesis (what we want to show)',\n            'Test statistic':'Measure of evidence against H₀',\n            'p-value':       'P(test stat as extreme or more | H₀ true)',\n            'α':             'Significance level (Type I error rate, usually 0.05)',\n            'Decision':      'Reject H₀ if p-value \u003C α'\n        },\n        'Error types': {\n            'Type I (α)':   'Reject H₀ when true (false positive)',\n            'Type II (β)':  'Fail to reject H₀ when false (false negative)',\n            'Power (1-β)':  'Probability of detecting true effect',\n            'Trade-off':    'Decreasing α increases β (for fixed n)'\n        },\n        'One-sample z-test': {\n            'H₀':           'μ = μ₀',\n            'Test stat':    'z = (x̄ - μ₀)\u002F(σ\u002F√n)',\n            'Use when':     'σ known, n large'\n        },\n        'One-sample t-test': {\n            'H₀':           'μ = μ₀',\n            'Test stat':    't = (x̄ - μ₀)\u002F(s\u002F√n)',\n            'df':           'n-1',\n            'Use when':     'σ unknown, normal population'\n        },\n        'Two-sample t-test': {\n            'H₀':           'μ₁ = μ₂',\n            'pooled':        't = (x̄₁-x̄₂)\u002F(sp√(1\u002Fn₁+1\u002Fn₂)), df=n₁+n₂-2',\n            'Welch':         't = (x̄₁-x̄₂)\u002F√(s₁²\u002Fn₁+s₂²\u002Fn₂), df=Satterthwaite',\n            'Use pooled when':'Equal variances (test with Levene\u002FF-test first)'\n        },\n        'Paired t-test': {\n            'H₀':           'μd = 0 (mean difference = 0)',\n            'Test stat':    't = d̄\u002F(sd\u002F√n), df = n-1',\n            'Use when':     'Matched pairs (before\u002Fafter, siblings)'\n        },\n        'Chi-square goodness of fit': {\n            'H₀':           'Data follows specified distribution',\n            'Test stat':    'χ² = Σ(O-E)²\u002FE',\n            'df':           'k-1-p (k categories, p estimated params)'\n        },\n        'Chi-square independence': {\n            'H₀':           'Two categorical variables independent',\n            'Test stat':    'χ² = Σ(Oᵢⱼ-Eᵢⱼ)²\u002FEᵢⱼ',\n            'Eᵢⱼ':          'Row total × Col total \u002F Grand total',\n            'df':           '(r-1)(c-1)'\n        }\n    }\n```\n\n---\n\n## Regression Analysis\n```python\ndef regression_models():\n    return {\n        'Simple Linear Regression': {\n            'model':        'Y = β₀ + β₁X + ε',\n            'OLS estimates':'β̂₁ = Sxy\u002FSxx, β̂₀ = ȳ - β̂₁x̄',\n            'Sxy':          'Σ(xᵢ-x̄)(yᵢ-ȳ)',\n            'Sxx':          'Σ(xᵢ-x̄)²',\n            'R²':           'SSR\u002FSST = 1 - SSE\u002FSST (proportion variance explained)',\n            'r':            '√R² × sign(β̂₁) = Pearson correlation',\n            'assumptions':  'LINEAR, INDEPENDENCE, NORMALITY, EQUAL VARIANCE (LINE)'\n        },\n        'Multiple Linear Regression': {\n            'model':        'Y = β₀ + β₁X₁ + ... + βₖXₖ + ε',\n            'matrix':       'β̂ = (XᵀX)⁻¹Xᵀy',\n            'R²':           'Always increases with more predictors',\n            'Adj R²':       '1-(1-R²)(n-1)\u002F(n-k-1) (penalizes extra vars)',\n            'F-test':       'Tests H₀: all β = 0 (overall significance)',\n            'VIF':          'Variance Inflation Factor (>10 = problematic multicollinearity)'\n        },\n        'Diagnostics': {\n            'Residual plots':  'vs fitted (homoscedasticity), vs order (independence)',\n            'QQ plot':         'Check normality of residuals',\n            'Leverage':        'hᵢᵢ = Xᵢ(XᵀX)⁻¹Xᵢᵀ (> 2k\u002Fn is high)',\n            'Cooks distance':  'Influence of single observation (>1 concerns)',\n            'DFFITS':          'Change in fit when obs removed'\n        },\n        'Logistic Regression': {\n            'model':        'log(p\u002F(1-p)) = β₀ + β₁X₁ + ...',\n            'logit':        'log(odds) = log(p\u002F(1-p))',\n            'probability':  'p = 1\u002F(1+e^(-Xβ))',\n            'fitting':      'Maximum likelihood (no closed form)',\n            'OR':           'Odds ratio = exp(βⱼ) per unit increase in Xⱼ',\n            'deviance':     '-2 log-likelihood (analogous to SSE)',\n            'pseudo-R²':    \"McFadden's: 1 - LL_full\u002FLL_null\"\n        },\n        'Model selection': {\n            'AIC':          '-2LL + 2k (lower is better)',\n            'BIC':          '-2LL + k·ln(n) (heavier penalty for complexity)',\n            'Cross-validation':'K-fold CV for prediction performance',\n            'Lasso':        'L1 penalty: β̂ = argmin(||y-Xβ||² + λΣ|βⱼ|) → sparse',\n            'Ridge':        'L2 penalty: β̂ = (XᵀX + λI)⁻¹Xᵀy → shrinkage'\n        }\n    }\n```\n\n---\n\n## ANOVA\n```\nOne-way ANOVA:\n  H₀: μ₁ = μ₂ = ... = μₖ (all group means equal)\n  Decomposition: SST = SSB + SSW\n    SST = Σᵢⱼ(yᵢⱼ - ȳ..)²  (total)\n    SSB = Σᵢ nᵢ(ȳᵢ. - ȳ..)²  (between groups)\n    SSW = Σᵢⱼ(yᵢⱼ - ȳᵢ.)²  (within groups\u002Ferror)\n\n  F = (SSB\u002F(k-1)) \u002F (SSW\u002F(N-k)) = MSB\u002FMSW\n  Under H₀: F ~ F(k-1, N-k)\n  Reject H₀ if F > F_{α, k-1, N-k}\n\nAssumptions:\n  Independence, normality (within groups), equal variances\n  Levene test for homogeneity of variance\n\nPost-hoc tests (if ANOVA significant):\n  Tukey HSD: controls family-wise error rate\n  Bonferroni: conservative, adjust α by number of tests\n  Scheffe: very conservative, flexible\n  Fisher LSD: liberal (no ANOVA protection needed)\n\nTwo-way ANOVA:\n  Factors A and B, interaction term A×B\n  Test: main effect A, main effect B, interaction A×B\n  If interaction significant: interpret with care\n\nEffect size:\n  η² = SSB\u002FSST  (eta-squared, proportion variance explained)\n  ω² = (SSB - (k-1)MSW)\u002F(SST + MSW)  (omega-squared, less biased)\n  Cohen's f: small=0.1, medium=0.25, large=0.4\n```\n\n---\n\n## Bayesian Statistics\n```python\ndef bayesian_framework():\n    return {\n        'Bayes Theorem': {\n            'formula':      'p(θ|data) ∝ p(data|θ) × p(θ)',\n            'posterior':    'p(θ|data): updated belief about θ',\n            'likelihood':   'p(data|θ): probability of data given θ',\n            'prior':        'p(θ): belief about θ before data',\n            'evidence':     'p(data): normalizing constant'\n        },\n        'Conjugate priors': {\n            'Beta-Binomial':    'Beta prior + Binomial likelihood = Beta posterior',\n            'Normal-Normal':    'Normal prior + Normal likelihood = Normal posterior',\n            'Gamma-Poisson':    'Gamma prior + Poisson likelihood = Gamma posterior',\n            'Dirichlet-Multinom':'Dirichlet prior + Multinomial = Dirichlet posterior',\n            'advantage':        'Closed-form posterior (no MCMC needed)'\n        },\n        'Prior selection': {\n            'Informative':  'Encodes strong prior knowledge',\n            'Weakly informative': 'Regularizes without strong beliefs (recommended)',\n            'Non-informative': 'Jeffreys prior: invariant to reparametrization',\n            'Flat':         'Uniform — NOT truly non-informative'\n        },\n        'Posterior summaries': {\n            'MAP':          'Maximum a posteriori: mode of posterior',\n            'Posterior mean':'E[θ|data]',\n            'Credible interval': '95% CI: P(a≤θ≤b|data) = 0.95 (TRUE probability!)',\n            'HDI':          'Highest density interval: shortest credible interval'\n        },\n        'MCMC (Markov Chain Monte Carlo)': {\n            'Metropolis-Hastings': 'Accept\u002Freject proposals',\n            'Gibbs sampling':      'Sample each parameter conditional on others',\n            'HMC':                 'Hamiltonian Monte Carlo: efficient for continuous',\n            'NUTS':                'No-U-Turn Sampler: adaptive HMC (Stan, PyMC)',\n            'Diagnostics':         'R̂ (\u003C 1.01), effective sample size, trace plots'\n        },\n        'Software': {\n            'Stan':     'Probabilistic programming, HMC\u002FNUTS, R or Python',\n            'PyMC':     'Python, NUTS, user-friendly',\n            'JAGS':     'Just Another Gibbs Sampler',\n            'brms':     'R, Bayesian regression using Stan'\n        }\n    }\n```\n\n---\n\n## Nonparametric Methods\n```\nWhen to use:\n  Non-normal data, small samples, ordinal data, robust analysis\n\nMann-Whitney U test:\n  Nonparametric alternative to two-sample t-test\n  Tests whether one distribution stochastically dominates\n  Uses ranks of combined sample\n\nWilcoxon signed-rank test:\n  Nonparametric alternative to paired t-test\n  Tests whether median difference = 0\n\nKruskal-Wallis test:\n  Nonparametric alternative to one-way ANOVA\n  Tests whether k groups have same distribution\n\nSpearman correlation:\n  rs = Pearson correlation of ranks\n  Robust to outliers, detects monotonic relationships\n  Test H₀: ρs = 0 using t-distribution\n\nBootstrap:\n  Resample with replacement from sample\n  Estimate sampling distribution empirically\n  Bootstrap CI: percentile, BCa, or t-bootstrap\n\nPermutation tests:\n  Permute labels\u002Fvalues, compute test statistic\n  p-value = proportion of permutations with stat ≥ observed\n  Exact test, no distributional assumptions\n```\n\n---\n\n## Power Analysis & Sample Size\n```python\ndef power_analysis():\n    return {\n        'Components': {\n            'α':        'Type I error rate (usually 0.05)',\n            'β':        'Type II error rate',\n            '1-β':      'Power (usually 0.80 or 0.90)',\n            'effect_size': 'Magnitude of difference to detect',\n            'n':        'Sample size (solve for this)'\n        },\n        'Effect sizes': {\n            'Cohens_d':     'd = (μ₁-μ₂)\u002Fσ  (small:0.2, medium:0.5, large:0.8)',\n            'Cohens_f':     'For ANOVA  (small:0.1, medium:0.25, large:0.4)',\n            'Cohens_w':     'For chi-square',\n            'r':            'Correlation  (small:0.1, medium:0.3, large:0.5)'\n        },\n        'Sample size for t-test': {\n            'formula':      'n = 2(z_α\u002F2 + z_β)²σ²\u002Fδ² per group',\n            'practical':    'Use G*Power, pwr package (R), scipy.stats.power'\n        },\n        'Multiple testing': {\n            'Problem':      'Testing m hypotheses: P(≥1 false positive) = 1-(1-α)ᵐ',\n            'Bonferroni':   'Reject if p \u003C α\u002Fm (controls FWER, conservative)',\n            'Holm':         'Stepdown Bonferroni (less conservative)',\n            'BH':           'Benjamini-Hochberg: controls FDR at level α',\n            'FDR':          'Expected proportion of false discoveries among rejections'\n        }\n    }\n```\n\n---\n\n## Common Pitfalls\n\n| Pitfall | Fix |\n|---|---|\n| p-value = probability H₀ true | p-value = P(data this extreme or more | H₀ true) |\n| Fail to reject = accept H₀ | Absence of evidence ≠ evidence of absence |\n| 95% CI contains parameter 95% of time | Correct! But not: 95% prob this specific CI contains it |\n| Correlation = causation | Correlation only; need experiments for causation |\n| Multiple testing ignored | Apply Bonferroni, BH, or Holm correction |\n| Normality assumed for large n | CLT justifies, but check for extreme outliers\u002Fskew |\n| Significant = important | Statistical significance ≠ practical significance (check effect size) |\n\n---\n\n## Related Skills\n\n- **probability-expert**: Deeper probability theory\n- **linear-algebra-expert**: Matrix algebra for regression\n- **machine-learning-expert**: Predictive modeling methods\n- **calculus-expert**: Mathematical foundations\n- **experimental-design-expert**: Study design principles\n- **data-science-expert**: Applied statistical analysis\n",{"name":5,"author":49,"license":15,"metadata":50,"description":7},"luo-kai",{"author":51,"version":52,"category":43},"luokai25","1.0","main","4d2fb89cdfb75f3f6fd3c6c5ec9ad90ea129eb3c","ai-agent-skills\u002F13-physics-and-mathematics (by Luo Kai)\u002Fstatistics-expert",2,"https:\u002F\u002Fgithub.com\u002Fluokai0\u002Fai-agent-skills-by-luo-kai","https:\u002F\u002Fgithub.com\u002Fluokai0","a solo builder.","2026-05-06 04:29:04",{"source":8,"skillName":5,"commands":62,"prompt":72,"zipAvailable":73},[63,66,69],{"tool":64,"command":65},"npx","npx skills add luokai0\u002Fai-agent-skills-by-luo-kai --skill statistics-expert",{"tool":67,"command":68},"bunx","bunx skills add luokai0\u002Fai-agent-skills-by-luo-kai --skill statistics-expert",{"tool":70,"command":71},"pnpm","pnpm dlx skills add luokai0\u002Fai-agent-skills-by-luo-kai --skill statistics-expert","安装技能，技能地址：https:\u002F\u002Fgithub.com\u002Fluokai0\u002Fai-agent-skills-by-luo-kai\u002Fblob\u002Fmain\u002Fai-agent-skills\u002F13-physics-and-mathematics (by Luo Kai)\u002Fstatistics-expert\u002FSKILL.md",true,[75,81],{"path":76,"resourceType":77,"resourceTypeDesc":78,"sizeBytes":79,"sourceHtmlUrl":80},".gitkeep","OTHER","其他",0,"https:\u002F\u002Fgithub.com\u002Fluokai0\u002Fai-agent-skills-by-luo-kai\u002Fblob\u002Fmain\u002Fai-agent-skills\u002F13-physics-and-mathematics (by Luo Kai)\u002Fstatistics-expert\u002F.gitkeep",{"path":82,"resourceType":83,"resourceTypeDesc":84,"sizeBytes":85,"sourceHtmlUrl":86},"minimax-skills.html","REFERENCE","参考文档",5327,"https:\u002F\u002Fgithub.com\u002FFeatherHunter\u002FSKILLS\u002Fblob\u002Fmain\u002Fminimax-skills\u002Fminimax-skills.html",[88,115,132,160,192,225],{"skillNo":89,"name":90,"nameZh":6,"description":91,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":92,"categories":93,"keywords":94,"publishedAt":44},"01M6F2SKTE208174C7S7JJCZC7","write-contracts","Generates secure Aptos Move V2 smart contracts with Object model, Digital Asset integration, security patterns, and storage type guidance. Includes comprehensive storage decision framework for optimal data structure selection. Triggers on: 'write contract', 'create NFT collection', 'build marketplace', 'implement minting', 'generate Move module', 'create token contract', 'build DAO', 'implement staking'. Ask storage questions when: 'store', 'track', 'registry', 'mapping', 'list', 'collection'.","https:\u002F\u002Fgithub.com\u002Fluokai0\u002Fai-agent-skills-by-luo-kai\u002Fblob\u002Fmain\u002Fai-agent-skills\u002F22-clawhub-skills\u002Fwrite-contracts\u002FSKILL.md",[],[95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114],"write","contracts","write contracts","storage","secure","create","generate","collection","model","decision","ask","smart-contracts","selection","objects","structure","nft","token","aptos","critical","move",{"skillNo":116,"name":117,"nameZh":6,"description":118,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":119,"categories":120,"keywords":121,"publishedAt":44},"01M6F2RFXJ6T9MKW0A7N7GM45B","web-search","通用网络搜索技能，支持多引擎搜索（百度、必应、DuckDuckGo），无需API密钥即可获取实时信息","https:\u002F\u002Fgithub.com\u002Fluokai0\u002Fai-agent-skills-by-luo-kai\u002Fblob\u002Fmain\u002Fai-agent-skills\u002F22-clawhub-skills\u002Fweb-search-ex-skill\u002FSKILL.md",[],[122,123,124,125,126,127,128,129,130,131],"web","search","web search","搜索","baidu","bing","duckduckgo","百度","多引擎","internet",{"skillNo":133,"name":134,"nameZh":6,"description":135,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":136,"categories":137,"keywords":141,"publishedAt":44},"01M6F2K968G1FZEZFPW2E6B222","solo-build","Execute implementation plan tasks with TDD workflow, auto-commit, and phase gates. Use when user says \"build it\", \"start building\", \"execute plan\", \"implement tasks\", \"ship it\", or references a track ID. Do NOT use for planning (use \u002Fplan) or scaffolding (use \u002Fscaffold).","https:\u002F\u002Fgithub.com\u002Fluokai0\u002Fai-agent-skills-by-luo-kai\u002Fblob\u002Fmain\u002Fai-agent-skills\u002F22-clawhub-skills\u002Fsolo-build\u002FSKILL.md",[138],{"categoryNo":139,"categoryName":140,"taxonomyType":21,"taxonomyTypeDesc":22},"01JBSEEDCAT000000000000023","Skill与智能体开发",[142,143,144,145,146,147,148,149,123,150,151,152,95,153,154,155,156,157,158,159],"build","solo","solo build","MCP","workflow","commit","tasks","code","Bash","plan","read","auto","references","context7","query","scaffolding","ship it","project",{"skillNo":161,"name":162,"nameZh":6,"description":163,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":164,"categories":165,"keywords":172,"publishedAt":44},"01M6F2GKN39SGHSPWYB3662YC8","react-email","Use when building HTML email templates with React components, adding a visual email editor to an application using the React Email visual editor, rendering emails to HTML, or sending emails with Resend. Covers welcome emails, password resets, notifications, order confirmations, newsletters, transactional emails, and the embeddable email editor component.","https:\u002F\u002Fgithub.com\u002Fluokai0\u002Fai-agent-skills-by-luo-kai\u002Fblob\u002Fmain\u002Fai-agent-skills\u002F22-clawhub-skills\u002Freact-email-skills\u002FSKILL.md",[166,169],{"categoryNo":167,"categoryName":168,"taxonomyType":21,"taxonomyTypeDesc":22},"01JBSEEDCAT000000000000005","邮件与沟通",{"categoryNo":170,"categoryName":171,"taxonomyType":21,"taxonomyTypeDesc":22},"01JBSEEDCAT000000000000009","前端开发",[173,174,175,176,177,178,179,180,181,182,183,184,185,40,186,187,188,189,190,191],"react","email","react email","github","editor","components","react components","component","welcome","https","notifications","password","node","html","resend","visual","emails","rendering","with",{"skillNo":193,"name":194,"nameZh":6,"description":195,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":196,"categories":197,"keywords":208,"publishedAt":44},"01M6F2FBVGVDFSG8SSFJGXA40C","APITester Agent-Driven API Testing","Test API endpoints and document responses. Define tests in plain English, run them, get formatted results. 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