bayesian-statistics-ai999x — offline, verified bayesian statistics solver with step-by-step answers

Install · Capability guide · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.0

Field: Mathematics & Statistics · Tasks: Question Answering, Sentence Similarity, Feature Extraction

bayesian-statistics-ai999x is an offline, deterministic Python solver that answers bayesian statistics questions with verified results from 78 certified capabilities, gives 378 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

bayesian-statistics-ai999x is a compact, deterministic bayesian statistics engine for Python. It reads a question in plain English (or a typed call), picks a certified bayesian statistics capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Highlights

• Verified, not generated. Every answer is computed by a certified bayesian statistics capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 378 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: Bayes' theorem, conjugate posteriors, credible intervals, Bayes factors, decision analysis, uncertainty propagation, Monte Carlo and sensitivity analysis.

Model overview

Property	Value
Package	bayesian-statistics-ai999x 1.0.0 (import bayesian_statistics_ai999x)
Type	Deterministic neuro-symbolic bayesian statistics engine (rule-based reading + certified symbolic capabilities)
Capabilities	78 certified procedures
Knowledge pack	version 1.0.1
Golden answers	378 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Quickstart

pip install bayesian-statistics-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/bayesian-statistics-ai999x/resolve/main/dist/bayesian_statistics_ai999x-1.0.0-py3-none-any.whl

import bayesian_statistics_ai999x as bayesian_statistics

bayesian_statistics.ask("A test has sensitivity 0.9 and specificity 0.95; the prevalence is 0.01. What is the positive predictive value?").show()
# -> SOLVED — ppv = 0.153846, npv = 0.998938, p_positive = 0.0585, false_discovery_rate = 0.846154  (diagnostic_test_predictive_values)
bayesian_statistics.ask("Beta-binomial posterior: prior Beta(2, 2), 7 successes in 10 trials. Find the posterior mean.").show()
# -> SOLVED — post_alpha = 9, post_beta = 5, posterior_mean = 0.642857, posterior_variance = 0.0153061  (beta_binomial_posterior)
bayesian_statistics.ask("95% highest-density interval of a Beta(3, 9) posterior.").show()
# -> SOLVED — lower = 0.0405552, upper = 0.483724, width = 0.443168  (beta_hdi)
bayesian_statistics.ask("Expected value of perfect information: payoffs = [[100, -20], [40, 30]], probabilities = [0.4, 0.6].").show()
# -> SOLVED — evpi = 24, ev_with_perfect_information = 58, best_expected_value = 34  (expected_value_of_perfect_information)
bayesian_statistics.solve("uncertainty_product_quotient", a=12.0, ua=0.3, b=4.0, ub=0.2).show()
# -> SOLVED — product = 48, u_product = 2.68328, quotient = 3, u_quotient = 0.167705  (uncertainty_product_quotient)
bayesian_statistics.ask("Who created you?").message
# -> bayesian-statistics-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

Command line: bayesian-statistics-ai999x "<question>", bayesian-statistics-ai999x --capabilities. The outputs above were produced by running this exact code on bayesian-statistics-ai999x 1.0.0.

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Capabilities

78 certified procedures. A few of them:

• Procedures: Bayes action: minimum posterior expected loss; Bayes factor for a normal mean (known variance); Bayes factor for a point null on a proportion (beta prior under H1); Bayes factor for two simple binomial hypotheses; Bayes' theorem for an event and its complement; Bayesian A/B test: probability that A beats B; Best action by expected monetary value; Beta prior from a mean and standard deviation; Beta-binomial conjugate posterior; Break-even (indifference) probability of two actions; Certainty equivalent with logarithmic utility; Control-variate Monte Carlo estimate; Covariance propagation through a linear map y = A x; Credible interval for a Poisson rate; Credible interval for a proportion (beta-binomial); Decision threshold probability (treat / do not treat); Decision under uncertainty: maximin, maximax, Laplace, minimax regret, Hurwicz; Dirichlet-multinomial conjugate posterior; Dirichlet-multinomial probability of a count vector (evidence); Effective degrees of freedom and expanded uncertainty (Welch-Satterthwaite); Equal-tailed credible interval of a beta distribution; Equal-tailed credible interval of a gamma distribution; Exact mean and standard deviation of a product of independent normals; Exact mean and variance of a polynomial of a normal variable …

bayesian_statistics_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 78 formulas

Show every formula, by topic

Bayes factors and model comparison

Bayes factor for a normal mean (known variance)

$$B{F}_{01}=\frac{p({\mu }_0\mid \bar{x},H_1)}{p({\mu }_0\mid H_1)}BF\_\{01\}\; =\; \backslash frac\{p(\backslash mu\_0\backslash mid\; \backslash bar\; x,\; H\_1)\}\{p(\backslash mu\_0\backslash mid\; H\_1)\}$$

Bayes factor for a point null on a proportion (beta prior under H1)

$$B{F}_{10}=\frac{B(\alpha +k,\beta +n-k)\mathrm{/}B(\alpha ,\beta )}{p_0^k(1-p_0{)}^{n-k}}BF\_\{10\}\; =\; \backslash frac\{B(\backslash alpha\; +\; k,\backslash \; \backslash beta\; +\; n\; -\; k)/B(\backslash alpha,\backslash beta)\}\{p\_0^\{k\}(1-p\_0)^\{n-k\}\}$$

Bayes factor for two simple binomial hypotheses

$$B{F}_{10}={\left(\frac{p_1}{p_0}\right)}^k{\left(\frac{1-p_1}{1-p_0}\right)}^{n-k}BF\_\{10\}\; =\; \backslash left(\backslash frac\{p\_1\}\{p\_0\}\backslash right)^\{k\}\backslash left(\backslash frac\{1-p\_1\}\{1-p\_0\}\backslash right)^\{n-k\}$$

Marginal likelihood (evidence) of binomial data under a beta prior

$$p(k)=\left(\genfrac{}{}{0px}{}{n}{k}\right)\frac{B(\alpha +k,\beta +n-k)}{B(\alpha ,\beta )}p(k)\; =\; \backslash binom\{n\}\{k\}\backslash frac\{B(\backslash alpha\; +\; k,\backslash \; \backslash beta\; +\; n\; -\; k)\}\{B(\backslash alpha,\backslash beta)\}$$

Posterior model probabilities from log evidences

$$P(M_i\mid D)=\frac{P(D\mid M_i)P(M_i)}{\sum _{j}P(D\mid M_j)P(M_j)}P(M\_i\backslash mid\; D)\; =\; \backslash frac\{P(D\backslash mid\; M\_i)P(M\_i)\}\{\backslash sum\_j\; P(D\backslash mid\; M\_j)P(M\_j)\}$$

Posterior probability of a hypothesis from a Bayes factor

$$\frac{P(H_1\mid D)}{P(H_0\mid D)}=B{F}_{10}\frac{P(H_1)}{P(H_0)}\backslash frac\{P(H\_1\backslash mid\; D)\}\{P(H\_0\backslash mid\; D)\}\; =\; BF\_\{10\}\backslash ,\backslash frac\{P(H\_1)\}\{P(H\_0)\}$$

Bayes' theorem and conditional probability

Bayes' theorem for an event and its complement

$$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B\mid A)P(A)+P(B\mid \mathrm{\neg }A)(1-P(A))}P(A\backslash mid\; B)\; =\; \backslash frac\{P(B\backslash mid\; A)\backslash ,P(A)\}\{P(B\backslash mid\; A)\backslash ,P(A)\; +\; P(B\backslash mid\; \backslash neg\; A)\backslash ,(1\; -\; P(A))\}$$

Law of total probability

$$P(B)=\sum _{i}P(B\mid A_i)P(A_i)P(B)\; =\; \backslash sum\_i\; P(B\backslash mid\; A\_i)\backslash ,P(A\_i)$$

Positive and negative predictive values of a test

$$\mathrm{P}\mathrm{P}\mathrm{V}=\frac{\mathrm{S}\mathrm{e}\pi }{\mathrm{S}\mathrm{e}\pi +(1-\mathrm{S}\mathrm{p})(1-\pi )},\mathrm{N}\mathrm{P}\mathrm{V}=\frac{\mathrm{S}\mathrm{p}(1-\pi )}{\mathrm{S}\mathrm{p}(1-\pi )+(1-\mathrm{S}\mathrm{e})\pi }\backslash mathrm\{PPV\}\; =\; \backslash frac\{\backslash mathrm\{Se\}\backslash ,\backslash pi\}\{\backslash mathrm\{Se\}\backslash ,\backslash pi\; +\; (1-\backslash mathrm\{Sp\})(1-\backslash pi)\},\backslash quad\; \backslash mathrm\{NPV\}\; =\; \backslash frac\{\backslash mathrm\{Sp\}\backslash ,(1-\backslash pi)\}\{\backslash mathrm\{Sp\}\backslash ,(1-\backslash pi)\; +\; (1-\backslash mathrm\{Se\})\backslash ,\backslash pi\}$$

Posterior after repeated independent tests

$${\text{odds}}_{post}=\frac{\pi }{1-\pi }(L{R}^{+}{)}^k(L{R}^{-}{)}^m\backslash text\{odds\}\_\{post\}\; =\; \backslash frac\{\backslash pi\}\{1-\backslash pi\}\backslash ,(LR^\{+\})^\{k\}\backslash ,(LR^\{-\})^\{m\}$$

Posterior over several discrete hypotheses

$$P(H_i\mid D)=\frac{P(D\mid H_i)P(H_i)}{\sum _{j}P(D\mid H_j)P(H_j)}P(H\_i\backslash mid\; D)\; =\; \backslash frac\{P(D\backslash mid\; H\_i)\backslash ,P(H\_i)\}\{\backslash sum\_j\; P(D\backslash mid\; H\_j)\backslash ,P(H\_j)\}$$

Monte Carlo estimation

Control-variate Monte Carlo estimate

$${\widehat{\mu }}_{cv}=\bar{y}-\widehat{b}(\bar{c}-{\mu }_c),\widehat{b}=\frac{\sum (y_i-\bar{y})(c_i-\bar{c})}{\sum (c_i-\bar{c}{)}^2}\backslash hat\backslash mu\_\{cv\}\; =\; \backslash bar\; y\; -\; \backslash hat\; b\backslash ,(\backslash bar\; c\; -\; \backslash mu\_c),\backslash quad\; \backslash hat\; b\; =\; \backslash frac\{\backslash sum\; (y\_i-\backslash bar\; y)(c\_i-\backslash bar\; c)\}\{\backslash sum\; (c\_i-\backslash bar\; c)^2\}$$

Importance sampling estimate and effective sample size

$$\widehat{\mu }=\frac{\sum _i{w}_{i}f(x_i)}{\sum _i{w}_i},w_i=\frac{p(x_i)}{q(x_i)},ESS=\frac{(\sum w_i{)}^2}{\sum w_i^2}\backslash hat\backslash mu\; =\; \backslash frac\{\backslash sum\_i\; w\_i\; f(x\_i)\}\{\backslash sum\_i\; w\_i\},\backslash \; w\_i\; =\; \backslash frac\{p(x\_i)\}\{q(x\_i)\},\backslash quad\; ESS\; =\; \backslash frac\{(\backslash sum\; w\_i)^2\}\{\backslash sum\; w\_i^2\}$$

Monte Carlo estimate of an integral from uniform samples

$$\widehat{I}=\frac{b-a}{n}\sum _{i}f(x_i),SE=(b-a)\frac{s_f}{\sqrt{n}}\backslash hat\; I\; =\; \backslash frac\{b-a\}\{n\}\backslash sum\_i\; f(x\_i),\backslash quad\; SE\; =\; (b-a)\backslash frac\{s\_f\}\{\backslash sqrt\; n\}$$

Number of Monte Carlo draws for a target accuracy

$$n=\lceil {\left(\frac{z_{(1+c)\mathrm{/}2}\sigma }{E}\right)}^2\rceil n\; =\; \backslash left\backslash lceil\backslash left(\backslash frac\{z\_\{(1+c)/2\}\backslash ,\backslash sigma\}\{E\}\backslash right)^2\backslash right\backslash rceil$$

Point of the Halton low-discrepancy sequence

$${\varphi }_b(i)=\sum _k{d}_k{b}^{-k-1}\text{ for }i=\sum _k{d}_k{b}^k\backslash phi\_b(i)\; =\; \backslash sum\_k\; d\_k\backslash ,b^\{-k-1\}\backslash \; \backslash text\{\; for\; \}\; i\; =\; \backslash sum\_k\; d\_k\; b^k$$

Stratified sampling estimate and standard error

$$\widehat{\mu }=\sum _h{W}_h{\bar{y}}_h,SE=\sqrt{\sum _h{W}_h^2{s}_h^2\mathrm{/}{n}_h}\backslash hat\backslash mu\; =\; \backslash sum\_h\; W\_h\backslash bar\; y\_h,\backslash quad\; SE\; =\; \backslash sqrt\{\backslash sum\_h\; W\_h^2\; s\_h^2/n\_h\}$$

Conjugate priors: Dirichlet-multinomial

Dirichlet-multinomial conjugate posterior

$$\theta \mid n\sim \mathrm{D}\mathrm{i}\mathrm{r}({\alpha }_1+n_1,\dots ,{\alpha }_k+n_k),E[{\theta }_j]=\frac{{\alpha }_j+n_j}{{\alpha }_0+n}\backslash theta\backslash mid\; n\; \backslash sim\; \backslash mathrm\{Dir\}(\backslash alpha\_1\; +\; n\_1,\; \backslash ldots,\; \backslash alpha\_k\; +\; n\_k),\backslash quad\; E[\backslash theta\_j]\; =\; \backslash frac\{\backslash alpha\_j\; +\; n\_j\}\{\backslash alpha\_0\; +\; n\}$$

Dirichlet-multinomial probability of a count vector (evidence)

$$P(n\mid \alpha )=\frac{n!\mathrm{\Gamma }({\alpha }_0)}{\mathrm{\Gamma }(n+{\alpha }_0)}\prod _j\frac{\mathrm{\Gamma }(n_j+{\alpha }_j)}{n_j!\mathrm{\Gamma }({\alpha }_j)}P(n\backslash mid\backslash alpha)\; =\; \backslash frac\{n!\backslash ,\backslash Gamma(\backslash alpha\_0)\}\{\backslash Gamma(n\; +\; \backslash alpha\_0)\}\backslash prod\_j\backslash frac\{\backslash Gamma(n\_j\; +\; \backslash alpha\_j)\}\{n\_j!\backslash ,\backslash Gamma(\backslash alpha\_j)\}$$

Conjugate priors: beta-binomial

Bayesian A/B test: probability that A beats B

$$P(p_B>p_A)=\sum _{i=0}^{{\alpha }_B-1}\frac{B({\alpha }_A+i,{\beta }_A+{\beta }_B)}{({\beta }_B+i)B(1+i,{\beta }_B)B({\alpha }_A,{\beta }_A)}P(p\_B\; >\; p\_A)\; =\; \backslash sum\_\{i=0\}^\{\backslash alpha\_B\; -\; 1\}\backslash frac\{B(\backslash alpha\_A\; +\; i,\backslash \; \backslash beta\_A\; +\; \backslash beta\_B)\}\{(\backslash beta\_B\; +\; i)\backslash ,B(1\; +\; i,\backslash \; \backslash beta\_B)\backslash ,B(\backslash alpha\_A,\; \backslash beta\_A)\}$$

Beta-binomial conjugate posterior

$$p\mid k\sim \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(\alpha +k,\beta +n-k),E[p\mid k]=\frac{\alpha +k}{\alpha +\beta +n}p\backslash mid\; k\; \backslash sim\; \backslash mathrm\{Beta\}(\backslash alpha\; +\; k,\backslash \; \backslash beta\; +\; n\; -\; k),\backslash quad\; E[p\backslash mid\; k]\; =\; \backslash frac\{\backslash alpha\; +\; k\}\{\backslash alpha\; +\; \backslash beta\; +\; n\}$$

Laplace's rule of succession

$$P(\text{next success})=\frac{k+1}{n+2}P(\backslash text\{next\; success\})\; =\; \backslash frac\{k+1\}\{n+2\}$$

MAP estimate of a proportion (beta-binomial)

$${\widehat{p}}_{MAP}=\frac{\alpha +k-1}{\alpha +\beta +n-2}\backslash hat\; p\_\{MAP\}\; =\; \backslash frac\{\backslash alpha\; +\; k\; -\; 1\}\{\backslash alpha\; +\; \backslash beta\; +\; n\; -\; 2\}$$

Posterior predictive probability (beta-binomial)

$$P(J=j)=\left(\genfrac{}{}{0px}{}{m}{j}\right)\frac{B(\alpha +j,\beta +m-j)}{B(\alpha ,\beta )}P(J\; =\; j)\; =\; \backslash binom\{m\}\{j\}\backslash frac\{B(\backslash alpha\; +\; j,\backslash \; \backslash beta\; +\; m\; -\; j)\}\{B(\backslash alpha,\; \backslash beta)\}$$

Posterior probability that p exceeds a value

$$P(p>c\mid k)=1-I_c(\alpha +k,\beta +n-k)P(p\; >\; c\; \backslash mid\; k)\; =\; 1\; -\; I\_c(\backslash alpha\; +\; k,\backslash \; \backslash beta\; +\; n\; -\; k)$$

Conjugate priors: gamma-Poisson and gamma-exponential

Gamma-Poisson conjugate posterior of a rate

$$\lambda \mid x\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}\text{\hspace{0.17em}⁣}(\alpha +\sum x_i,\beta +n)\backslash lambda\backslash mid\; x\; \backslash sim\; \backslash mathrm\{Gamma\}\backslash !\backslash left(\backslash alpha\; +\; \backslash textstyle\backslash sum\; x\_i,\backslash \; \backslash beta\; +\; n\backslash right)$$

Gamma-exponential conjugate posterior of a rate

$$\lambda \mid t\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}\text{\hspace{0.17em}⁣}(\alpha +n,\beta +\sum t_i)\backslash lambda\backslash mid\; t\; \backslash sim\; \backslash mathrm\{Gamma\}\backslash !\backslash left(\backslash alpha\; +\; n,\backslash \; \backslash beta\; +\; \backslash textstyle\backslash sum\; t\_i\backslash right)$$

MAP estimate of a Poisson rate (gamma prior)

$${\widehat{\lambda }}_{MAP}=\frac{\alpha +\sum x_i-1}{\beta +n}\backslash hat\backslash lambda\_\{MAP\}\; =\; \backslash frac\{\backslash alpha\; +\; \backslash sum\; x\_i\; -\; 1\}\{\backslash beta\; +\; n\}$$

Posterior predictive count (negative binomial)

$$P(X=k)=\frac{\mathrm{\Gamma }(\alpha +k)}{\mathrm{\Gamma }(\alpha )k!}{\left(\frac{\beta }{\beta +1}\right)}^{\alpha }{\left(\frac{1}{\beta +1}\right)}^{k}P(X\; =\; k)\; =\; \backslash frac\{\backslash Gamma(\backslash alpha\; +\; k)\}\{\backslash Gamma(\backslash alpha)\backslash ,k!\}\backslash left(\backslash frac\{\backslash beta\}\{\backslash beta+1\}\backslash right)^\{\backslash alpha\}\backslash left(\backslash frac\{1\}\{\backslash beta+1\}\backslash right)^\{k\}$$

Probability that one Poisson rate exceeds another (gamma posteriors)

$$P({\lambda }_1>{\lambda }_2)=1-I_{b_1\mathrm{/}(b_1+b_2)}(a_1,a_2)P(\backslash lambda\_1\; >\; \backslash lambda\_2)\; =\; 1\; -\; I\_\{b\_1/(b\_1+b\_2)\}(a\_1,\; a\_2)$$

Conjugate priors: normal models

Normal mean with known variance: posterior and credible interval from a sample mean

$${\mu }_n=B{\mu }_0+(1-B)\bar{x},B=\frac{{\sigma }^2\mathrm{/}n}{{\sigma }^2\mathrm{/}n+{\tau }_0^2},{\mu }_n\pm z_{(1+c)\mathrm{/}2}{\tau }_n\backslash mu\_n\; =\; B\backslash mu\_0\; +\; (1-B)\backslash bar\; x,\backslash \; B\; =\; \backslash frac\{\backslash sigma^2/n\}\{\backslash sigma^2/n\; +\; \backslash tau\_0^2\},\backslash quad\; \backslash mu\_n\; \backslash pm\; z\_\{(1+c)/2\}\backslash ,\backslash tau\_n$$

Normal mean with known variance: posterior from data

$$\frac{1}{{\tau }_n^2}=\frac{1}{{\tau }_0^2}+\frac{n}{{\sigma }^2},{\mu }_n={\tau }_n^2(\frac{{\mu }_0}{{\tau }_0^2}+\frac{n\bar{x}}{{\sigma }^2})\backslash frac\{1\}\{\backslash tau\_n^2\}\; =\; \backslash frac\{1\}\{\backslash tau\_0^2\}\; +\; \backslash frac\{n\}\{\backslash sigma^2\},\backslash quad\; \backslash mu\_n\; =\; \backslash tau\_n^2\backslash left(\backslash frac\{\backslash mu\_0\}\{\backslash tau\_0^2\}\; +\; \backslash frac\{n\backslash bar\; x\}\{\backslash sigma^2\}\backslash right)$$

Normal model with unknown mean and variance (normal-inverse-gamma)

$${\beta }_n={\beta }_0+\frac{1}{2}\sum (x_i-\bar{x}{)}^2+\frac{{\kappa }_{0}n(\bar{x}-{\mu }_0{)}^2}{2({\kappa }_0+n)},\mu \mid x\sim t_{2{\alpha }_n}\text{\hspace{0.17em}⁣}({\mu }_n,\frac{{\beta }_n}{{\alpha }_n{\kappa }_n})\backslash beta\_n\; =\; \backslash beta\_0\; +\; \backslash tfrac12\backslash sum(x\_i-\backslash bar\; x)^2\; +\; \backslash frac\{\backslash kappa\_0\; n(\backslash bar\; x-\backslash mu\_0)^2\}\{2(\backslash kappa\_0+n)\},\backslash quad\; \backslash mu\backslash mid\; x\; \backslash sim\; t\_\{2\backslash alpha\_n\}\backslash !\backslash left(\backslash mu\_n,\backslash \; \backslash frac\{\backslash beta\_n\}\{\backslash alpha\_n\backslash kappa\_n\}\backslash right)$$

Posterior predictive distribution of a new observation

$$Y_{new}\mid \bar{x}\sim N\text{\hspace{0.17em}⁣}({\mu }_n,{\tau }_n^2+{\sigma }^2)Y\_\{new\}\backslash mid\; \backslash bar\; x\; \backslash sim\; N\backslash !\backslash left(\backslash mu\_n,\backslash \; \backslash tau\_n^2\; +\; \backslash sigma^2\backslash right)$$

Credible intervals

Credible interval for a Poisson rate

$$[\frac{G_{{\alpha }^{\mathrm{\prime }}}^{-1}\text{\hspace{0.17em}⁣}\left(\frac{1-c}{2}\right)}{{\beta }^{\mathrm{\prime }}},\frac{G_{{\alpha }^{\mathrm{\prime }}}^{-1}\text{\hspace{0.17em}⁣}\left(\frac{1+c}{2}\right)}{{\beta }^{\mathrm{\prime }}}]\backslash left[\backslash frac\{G^\{-1\}\_\{\backslash alpha\text{'}\}\backslash !\backslash left(\backslash frac\{1-c\}\{2\}\backslash right)\}\{\backslash beta\text{'}\},\backslash \; \backslash frac\{G^\{-1\}\_\{\backslash alpha\text{'}\}\backslash !\backslash left(\backslash frac\{1+c\}\{2\}\backslash right)\}\{\backslash beta\text{'}\}\backslash right]$$

Credible interval for a proportion (beta-binomial)

$$p\mid k\sim \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(\alpha +k,\beta +n-k),[q_{(1-c)\mathrm{/}2},q_{(1+c)\mathrm{/}2}]p\backslash mid\; k\; \backslash sim\; \backslash mathrm\{Beta\}(\backslash alpha+k,\backslash beta+n-k),\backslash quad\; [q\_\{(1-c)/2\},\backslash \; q\_\{(1+c)/2\}]$$

Equal-tailed credible interval of a beta distribution

$$[F_{\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(\alpha ,\beta )}^{-1}\text{\hspace{0.17em}⁣}\left(\frac{1-c}{2}\right),F_{\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(\alpha ,\beta )}^{-1}\text{\hspace{0.17em}⁣}\left(\frac{1+c}{2}\right)]\backslash left[F^\{-1\}\_\{\backslash mathrm\{Beta\}(\backslash alpha,\backslash beta)\}\backslash !\backslash left(\backslash tfrac\{1-c\}\{2\}\backslash right),\backslash \; F^\{-1\}\_\{\backslash mathrm\{Beta\}(\backslash alpha,\backslash beta)\}\backslash !\backslash left(\backslash tfrac\{1+c\}\{2\}\backslash right)\backslash right]$$

Equal-tailed credible interval of a gamma distribution

$$[G^{-1}\text{\hspace{0.17em}⁣}\left(\frac{1-c}{2}\right),G^{-1}\text{\hspace{0.17em}⁣}\left(\frac{1+c}{2}\right)]\backslash left[G^\{-1\}\backslash !\backslash left(\backslash tfrac\{1-c\}\{2\}\backslash right),\backslash \; G^\{-1\}\backslash !\backslash left(\backslash tfrac\{1+c\}\{2\}\backslash right)\backslash right]$$

Highest-density interval of a beta posterior

$$F(u)-F(l)=c,f(l)=f(u)F(u)\; -\; F(l)\; =\; c,\backslash quad\; f(l)\; =\; f(u)$$

Highest-density interval of a gamma posterior

$$G(u)-G(l)=c,g(l)=g(u)G(u)\; -\; G(l)\; =\; c,\backslash quad\; g(l)\; =\; g(u)$$

Jeffreys interval for a binomial proportion

$$[B^{-1}\text{\hspace{0.17em}⁣}(\frac{1-c}{2};k+\frac{1}{2},n-k+\frac{1}{2}),B^{-1}\text{\hspace{0.17em}⁣}(\frac{1+c}{2};k+\frac{1}{2},n-k+\frac{1}{2})]\backslash left[B^\{-1\}\backslash !\backslash left(\backslash tfrac\{1-c\}\{2\};\backslash ,k+\backslash tfrac12,\backslash ,n-k+\backslash tfrac12\backslash right),\backslash \; B^\{-1\}\backslash !\backslash left(\backslash tfrac\{1+c\}\{2\};\backslash ,k+\backslash tfrac12,\backslash ,n-k+\backslash tfrac12\backslash right)\backslash right]$$

Decision analysis

Bayes action: minimum posterior expected loss

$$\rho (a)=\sum _{s}P(s\mid x)L(a,s),a^{\ast }=\arg \underset{a}{\min }\rho (a)\backslash rho(a)\; =\; \backslash sum\_s\; P(s\backslash mid\; x)\backslash ,L(a,s),\backslash quad\; a^*\; =\; \backslash arg\backslash min\_a\; \backslash rho(a)$$

Best action by expected monetary value

$$EV(a)=\sum _{s}P(s)u(a,s),a^{\ast }=\arg \underset{a}{\max }EV(a)EV(a)\; =\; \backslash sum\_s\; P(s)\backslash ,u(a,\; s),\backslash quad\; a^*\; =\; \backslash arg\backslash max\_a\; EV(a)$$

Break-even (indifference) probability of two actions

$$p^{\ast }=\frac{u_{22}-u_{12}}{(u_{11}-u_{12})-(u_{21}-u_{22})}p^*\; =\; \backslash frac\{u\_\{22\}\; -\; u\_\{12\}\}\{(u\_\{11\}\; -\; u\_\{12\})\; -\; (u\_\{21\}\; -\; u\_\{22\})\}$$

Certainty equivalent with logarithmic utility

$$CE=\exp \text{\hspace{0.17em}⁣}(\sum _i{p}_i\ln x_i)=\prod _i{x}_i^{p_i}CE\; =\; \backslash exp\backslash !\backslash left(\backslash sum\_i\; p\_i\; \backslash ln\; x\_i\backslash right)\; =\; \backslash prod\_i\; x\_i^\{p\_i\}$$

Decision threshold probability (treat / do not treat)

$$p^{\ast }=\frac{L_{FP}}{L_{FP}+L_{FN}}p^*\; =\; \backslash frac\{L\_\{FP\}\}\{L\_\{FP\}\; +\; L\_\{FN\}\}$$

Decision under uncertainty: maximin, maximax, Laplace, minimax regret, Hurwicz

$$\text{maximin: }\underset{a}{\max }\underset{s}{\min }u,\text{regret }r(a,s)=\underset{a^{\mathrm{\prime }}}{\max }u(a^{\mathrm{\prime }},s)-u(a,s)\backslash text\{maximin:\; \}\backslash max\_a\backslash min\_s\; u,\backslash quad\; \backslash text\{regret\; \}\; r(a,s)\; =\; \backslash max\_\{a\text{'}\}u(a\text{'},s)\; -\; u(a,s)$$

Expected utility and certainty equivalent (exponential utility)

$$CE=-R\ln \sum _i{p}_i{e}^{-x_i\mathrm{/}R}CE\; =\; -R\backslash ln\backslash sum\_i\; p\_i\; e^\{-x\_i/R\}$$

Expected value of perfect information (EVPI)

$$EVPI=\sum _{s}P(s)\underset{a}{\max }u(a,s)-\underset{a}{\max }\sum _{s}P(s)u(a,s)EVPI\; =\; \backslash sum\_s\; P(s)\backslash max\_a\; u(a,s)\; -\; \backslash max\_a\backslash sum\_s\; P(s)\backslash ,u(a,s)$$

Expected value of sample information (EVSI)

$$EVSI=\sum _{z}P(z)\underset{a}{\max }\sum _{s}P(s\mid z)u(a,s)-\underset{a}{\max }\sum _{s}P(s)u(a,s)EVSI\; =\; \backslash sum\_z\; P(z)\backslash max\_a\backslash sum\_s\; P(s\backslash mid\; z)\backslash ,u(a,s)\; -\; \backslash max\_a\backslash sum\_s\; P(s)\backslash ,u(a,s)$$

Information and quadrature for Bayesian computation

Gauss-Hermite rule for expectations under a normal distribution

$$E[f(Z)]\approx \sum _{i=1}^n{w}_{i}f(x_i),H{e}_n(x_i)=0,w_i=\frac{n!}{n^{2}H{e}_{n-1}(x_i{)}^2}E[f(Z)]\; \backslash approx\; \backslash sum\_\{i=1\}^\{n\}\; w\_i\; f(x\_i),\backslash quad\; He\_n(x\_i)\; =\; 0,\backslash quad\; w\_i\; =\; \backslash frac\{n!\}\{n^2\backslash ,He\_\{n-1\}(x\_i)^2\}$$

Kullback-Leibler divergence between two beta distributions

$$KL=\ln \frac{B({\alpha }_2,{\beta }_2)}{B({\alpha }_1,{\beta }_1)}+({\alpha }_1-{\alpha }_2)\psi ({\alpha }_1)+({\beta }_1-{\beta }_2)\psi ({\beta }_1)+({\alpha }_2-{\alpha }_1+{\beta }_2-{\beta }_1)\psi ({\alpha }_1+{\beta }_1)KL\; =\; \backslash ln\backslash frac\{B(\backslash alpha\_2,\backslash beta\_2)\}\{B(\backslash alpha\_1,\backslash beta\_1)\}\; +\; (\backslash alpha\_1-\backslash alpha\_2)\backslash psi(\backslash alpha\_1)\; +\; (\backslash beta\_1-\backslash beta\_2)\backslash psi(\backslash beta\_1)\; +\; (\backslash alpha\_2-\backslash alpha\_1+\backslash beta\_2-\backslash beta\_1)\backslash psi(\backslash alpha\_1+\backslash beta\_1)$$

Kullback-Leibler divergence between two normal distributions

$$KL(N_1\mathrm{\parallel }{N}_2)=\ln \frac{{\sigma }_2}{{\sigma }_1}+\frac{{\sigma }_1^2+({\mu }_1-{\mu }_2{)}^2}{2{\sigma }_2^2}-\frac{1}{2}KL(N\_1\backslash Vert\; N\_2)\; =\; \backslash ln\backslash frac\{\backslash sigma\_2\}\{\backslash sigma\_1\}\; +\; \backslash frac\{\backslash sigma\_1^2\; +\; (\backslash mu\_1-\backslash mu\_2)^2\}\{2\backslash sigma\_2^2\}\; -\; \backslash frac12$$

Odds and likelihood ratios

Likelihood ratios of a diagnostic test

$$L{R}^{+}=\frac{\mathrm{S}\mathrm{e}}{1-\mathrm{S}\mathrm{p}},L{R}^{-}=\frac{1-\mathrm{S}\mathrm{e}}{\mathrm{S}\mathrm{p}}LR^\{+\}\; =\; \backslash frac\{\backslash mathrm\{Se\}\}\{1-\backslash mathrm\{Sp\}\},\backslash quad\; LR^\{-\}\; =\; \backslash frac\{1-\backslash mathrm\{Se\}\}\{\backslash mathrm\{Sp\}\}$$

Odds and log-odds of a probability

$$\text{odds}=\frac{p}{1-p},\mathrm{logit}p=\ln \frac{p}{1-p}\backslash text\{odds\}\; =\; \backslash frac\{p\}\{1-p\},\backslash quad\; \backslash operatorname\{logit\}\; p\; =\; \backslash ln\backslash frac\{p\}\{1-p\}$$

Post-test probability from a likelihood ratio

$$\frac{P(H\mid E)}{1-P(H\mid E)}=\frac{P(H)}{1-P(H)}\times LR\backslash frac\{P(H\backslash mid\; E)\}\{1-P(H\backslash mid\; E)\}\; =\; \backslash frac\{P(H)\}\{1-P(H)\}\backslash times\; LR$$

Probability from odds

$$p=\frac{o}{1+o}p\; =\; \backslash frac\{o\}\{1+o\}$$

Posterior approximation

Laplace (normal) approximation of a beta posterior

$$\widehat{p}=\frac{\alpha -1}{\alpha +\beta -2},{\sigma }^2={[\frac{\alpha -1}{{\widehat{p}}^2}+\frac{\beta -1}{(1-\widehat{p}{)}^2}]}^{-1}\backslash hat\; p\; =\; \backslash frac\{\backslash alpha-1\}\{\backslash alpha+\backslash beta-2\},\backslash quad\; \backslash sigma^2\; =\; \backslash left[\backslash frac\{\backslash alpha-1\}\{\backslash hat\; p^2\}\; +\; \backslash frac\{\backslash beta-1\}\{(1-\backslash hat\; p)^2\}\backslash right]^\{-1\}$$

Prior elicitation

Beta prior from a mean and standard deviation

$$\nu =\frac{m(1-m)}{s^2}-1,\alpha =m\nu ,\beta =(1-m)\nu \backslash nu\; =\; \backslash frac\{m(1-m)\}\{s^2\}\; -\; 1,\backslash quad\; \backslash alpha\; =\; m\backslash nu,\backslash quad\; \backslash beta\; =\; (1-m)\backslash nu$$

Sensitivity analysis

Morris elementary-effects screening

$$E{E}_j=\frac{f(x+\mathrm{\Delta }{e}_j)-f(x)}{\mathrm{\Delta }},{\mu }_j^{\ast }=\frac{1}{r}\sum _t\mathrm{\mid }E{E}_j^{(t)}\mathrm{\mid }EE\_j\; =\; \backslash frac\{f(x\; +\; \backslash Delta\; e\_j)\; -\; f(x)\}\{\backslash Delta\},\backslash quad\; \backslash mu\_j^*\; =\; \backslash frac1r\backslash sum\_t\; \backslash vert\; EE\_j^\{(t)\}\backslash vert$$

Sobol indices of Sobol's G-function

$$V_i=\frac{1}{3(1+a_i{)}^2},V=\prod _i(1+V_i)-1,S_i=\frac{V_i}{V},S_{Ti}=\frac{V_i\prod _{j\mathrm{\ne }i}(1+V_j)}{V}V\_i\; =\; \backslash frac\{1\}\{3(1+a\_i)^2\},\backslash \; V\; =\; \backslash prod\_i(1+V\_i)\; -\; 1,\backslash \; S\_i\; =\; \backslash frac\{V\_i\}\{V\},\backslash \; S\_\{Ti\}\; =\; \backslash frac\{V\_i\backslash prod\_\{j\backslash ne\; i\}(1+V\_j)\}\{V\}$$

Sobol indices of a linear (additive) model

$$S_i=\frac{a_i^2{\sigma }_i^2}{\sum _j{a}_j^2{\sigma }_j^2}S\_i\; =\; \backslash frac\{a\_i^2\backslash sigma\_i^2\}\{\backslash sum\_j\; a\_j^2\backslash sigma\_j^2\}$$

Sobol indices of a model with an interaction term

$$V_1=(a_1+a_{12}{\mu }_2{)}^2{\sigma }_1^2,V_2=(a_2+a_{12}{\mu }_1{)}^2{\sigma }_2^2,V_{12}=a_{12}^2{\sigma }_1^2{\sigma }_2^{2}V\_1\; =\; (a\_1\; +\; a\_\{12\}\backslash mu\_2)^2\backslash sigma\_1^2,\backslash \; V\_2\; =\; (a\_2\; +\; a\_\{12\}\backslash mu\_1)^2\backslash sigma\_2^2,\backslash \; V\_\{12\}\; =\; a\_\{12\}^2\backslash sigma\_1^2\backslash sigma\_2^2$$

Sobol sensitivity indices of the Ishigami function

$$V=\frac{a^2}{8}+\frac{b{\pi }^4}{5}+\frac{b^2{\pi }^8}{18}+\frac{1}{2},V_1=\frac{1}{2}{(1+\frac{b{\pi }^4}{5})}^2,V_2=\frac{a^2}{8},V_{13}=\frac{8b^2{\pi }^8}{225}V\; =\; \backslash frac\{a^2\}\{8\}\; +\; \backslash frac\{b\backslash pi^4\}\{5\}\; +\; \backslash frac\{b^2\backslash pi^8\}\{18\}\; +\; \backslash frac12,\backslash \; V\_1\; =\; \backslash frac12\backslash left(1\; +\; \backslash frac\{b\backslash pi^4\}\{5\}\backslash right)^2,\backslash \; V\_2\; =\; \backslash frac\{a^2\}\{8\},\backslash \; V\_\{13\}\; =\; \backslash frac\{8b^2\backslash pi^8\}\{225\}$$

Uncertainty propagation

Covariance propagation through a linear map y = A x

$${\mathrm{\Sigma }}_y=A{\mathrm{\Sigma }}_x{A}^{\mathsf{T}}\backslash Sigma\_y\; =\; A\backslash ,\backslash Sigma\_x\backslash ,A^\{\backslash mathsf\; T\}$$

Effective degrees of freedom and expanded uncertainty (Welch-Satterthwaite)

$${\nu }_{eff}=\frac{u_c^4}{\sum _i{u}_i^4\mathrm{/}{\nu }_i},U=t_{(1+p)\mathrm{/}2,{\nu }_{eff}}{u}_c\backslash nu\_\{eff\}\; =\; \backslash frac\{u\_c^4\}\{\backslash sum\_i\; u\_i^4/\backslash nu\_i\},\backslash quad\; U\; =\; t\_\{(1+p)/2,\backslash ,\backslash nu\_\{eff\}\}\backslash ,u\_c$$

Exact mean and standard deviation of a product of independent normals

$$\mathrm{V}\mathrm{a}\mathrm{r}(XY)={\mu }_x^2{\sigma }_y^2+{\mu }_y^2{\sigma }_x^2+{\sigma }_x^2{\sigma }_y^2\backslash mathrm\{Var\}(XY)\; =\; \backslash mu\_x^2\backslash sigma\_y^2\; +\; \backslash mu\_y^2\backslash sigma\_x^2\; +\; \backslash sigma\_x^2\backslash sigma\_y^2$$

Exact mean and variance of a polynomial of a normal variable

$$E[Z^{2m}]=(2m-1)!!,X=\mu +\sigma ZE[Z^\{2m\}]\; =\; (2m-1)!!,\backslash quad\; X\; =\; \backslash mu\; +\; \backslash sigma\; Z$$

Exact versus first-order propagation through exp(X)

$$E[e^X]=e^{\mu +{\sigma }^2\mathrm{/}2},\mathrm{S}\mathrm{D}[e^X]=e^{\mu +{\sigma }^2\mathrm{/}2}\sqrt{e^{{\sigma }^2}-1}E[e^X]\; =\; e^\{\backslash mu\; +\; \backslash sigma^2/2\},\backslash quad\; \backslash mathrm\{SD\}[e^X]\; =\; e^\{\backslash mu\; +\; \backslash sigma^2/2\}\backslash sqrt\{e^\{\backslash sigma^2\}\; -\; 1\}$$

Inverse-variance weighted mean of measurements

$${\bar{x}}_w=\frac{\sum x_i\mathrm{/}{u}_i^2}{\sum 1\mathrm{/}{u}_i^2},u({\bar{x}}_w)=(\sum 1\mathrm{/}{u}_i^2{)}^{-1\mathrm{/}2}\backslash bar\; x\_w\; =\; \backslash frac\{\backslash sum\; x\_i/u\_i^2\}\{\backslash sum\; 1/u\_i^2\},\backslash quad\; u(\backslash bar\; x\_w)\; =\; \backslash Big(\backslash sum\; 1/u\_i^2\backslash Big)^\{-1/2\}$$

Type B standard uncertainty from a stated half-width

$$u=\frac{a}{\sqrt{3}}(\text{rect.}),\frac{a}{\sqrt{6}}(\text{tri.}),\frac{a}{\sqrt{2}}(\text{arcsine})u\; =\; \backslash frac\{a\}\{\backslash sqrt\; 3\}\backslash \; (\backslash text\{rect.\}),\backslash \; \backslash frac\{a\}\{\backslash sqrt\; 6\}\backslash \; (\backslash text\{tri.\}),\backslash \; \backslash frac\{a\}\{\backslash sqrt\; 2\}\backslash \; (\backslash text\{arcsine\})$$

Uncertainty of a function of one variable

$$u_f=\mid f^{\mathrm{\prime }}(x)\mid u_{x}u\_f\; =\; \backslash left\backslash vert\; f\text{'}(x)\backslash right\backslash vert\; u\_x$$

Uncertainty of a linear combination of correlated inputs

$$u_y^2=\sum _i\sum _j{c}_i{c}_j{u}_i{u}_j{r}_{ij}u\_y^2\; =\; \backslash sum\_i\backslash sum\_j\; c\_i\; c\_j\; u\_i\; u\_j\; r\_\{ij\}$$

Uncertainty of a power x^n

$$\frac{u_y}{\mathrm{\mid }y\mathrm{\mid }}=\mathrm{\mid }n\mathrm{\mid }\frac{u_x}{\mathrm{\mid }x\mathrm{\mid }}\backslash frac\{u\_y\}\{\backslash vert\; y\backslash vert\; \}\; =\; \backslash vert\; n\backslash vert\; \backslash ,\backslash frac\{u\_x\}\{\backslash vert\; x\backslash vert\; \}$$

Uncertainty of a power-law product y = k x1^p1 x2^p2 ...

$$\frac{u_y}{\mathrm{\mid }y\mathrm{\mid }}=\sqrt{\sum _i{\left(p_i\frac{u_i}{x_i}\right)}^2}\backslash frac\{u\_y\}\{\backslash vert\; y\backslash vert\; \}\; =\; \backslash sqrt\{\backslash sum\_i\; \backslash left(p\_i\backslash ,\backslash frac\{u\_i\}\{x\_i\}\backslash right)^2\}$$

Uncertainty of a product and a quotient

$$\frac{u_{ab}}{\mathrm{\mid }ab\mathrm{\mid }}=\frac{u_{a\mathrm{/}b}}{\mathrm{\mid }a\mathrm{/}b\mathrm{\mid }}=\sqrt{{\left(\frac{u_a}{a}\right)}^2+{\left(\frac{u_b}{b}\right)}^2}\backslash frac\{u\_\{ab\}\}\{\backslash vert\; ab\backslash vert\; \}\; =\; \backslash frac\{u\_\{a/b\}\}\{\backslash vert\; a/b\backslash vert\; \}\; =\; \backslash sqrt\{\backslash left(\backslash frac\{u\_a\}\{a\}\backslash right)^2\; +\; \backslash left(\backslash frac\{u\_b\}\{b\}\backslash right)^2\}$$

Uncertainty of a product and quotient with correlation

$$u_{ab}^2=b^2{u}_a^2+a^2{u}_b^2+2ab\rho u_a{u}_{b}u\_\{ab\}^2\; =\; b^2u\_a^2\; +\; a^2u\_b^2\; +\; 2ab\backslash ,\backslash rho\backslash ,u\_au\_b$$

Uncertainty of a sum or linear combination (independent inputs)

$$u_y=\sqrt{\sum _i{c}_i^2{u}_i^2}u\_y\; =\; \backslash sqrt\{\backslash sum\_i\; c\_i^2\; u\_i^2\}$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified bayesian statistics capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (Bayes' theorem, conjugate posteriors, credible intervals, Bayes factors, decision analysis, uncertainty propagation, Monte Carlo and sensitivity analysis); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{bayesian_statistics_ai999x_1_0_0,
  title  = {bayesian-statistics-ai999x: an offline, verified bayesian statistics solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.0},
  url    = {https://huggingface.co/AI999X/bayesian-statistics-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/bayesian-statistics-ai999x/discussions or visit https://ai999x.com.

Support the project

bayesian-statistics-ai999x is free and open source. If it helps you, you can support further development on Buy Me a Coffee.

Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com