ToFUL — Tool for Uncertainty Learning

An interactive web application and Python computation engine for computing statistical moments of probability distributions.

$$\mu_r(a) = E\left[(X - a)^r\right]$$

Compute raw moments, central moments, variance, skewness, kurtosis, and more for both discrete and continuous random variables — with rigorous numerical methods, convergence acceleration, and real-time validation.

🚀 Try it now — no installation needed
📖 Read the documentation

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Features

Feature	Details
Discrete RVs	Finite and infinite support, with 5-method convergence cascade
Continuous RVs	Domain-aware quadrature: Gauss-Laguerre, Gauss-Hermite, tanh-sinh
Exact results	SymPy symbolic integration for distributions without conditionals
High precision	mpmath arbitrary-precision mode above 12 decimal places
Parallel computation	All moment orders computed concurrently via ThreadPoolExecutor
Smart input	Auto-corrects ^, `
Live validation	PMF sum and PDF integral checked before any moment is computed
5 result tabs	Moments · Statistics · Distribution plot · Table (CSV) · Convergence

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Quick Start

Online (recommended)

Visit toful1.streamlit.app — no installation needed.

Local installation

git clone https://github.com/pranava-ba/ToFUL.git
cd ToFUL
pip install -r requirements.txt
streamlit run app.py

All three files — app.py, core.py, and toful_parser.py — must be in the same directory.

Verify the installation

python -c "
import core, toful_parser, numpy as np

x, _, _ = core.parse_range_input('0,1,2,3,...')
vr = core.validate_drv_probabilities(
    '0.3*(0.7**x) if x>=0 else 0', x[:200], True
)
print('Valid:', vr.is_valid, '| Sum:', round(vr.integral_or_sum, 6))
"
# Valid: True | Sum: 1.0

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Usage Examples

Discrete — Geometric Distribution

Field	Value
Variable type	Discrete (DRV)
Range	0,1,2,3,...
PMF	0.3 * (0.7 ** x) if x >= 0 else 0
Moment reference	Origin (a = 0)
Max order	4

Expected output:

Moment	Value	Method
μ₁	2.333333	term-magnitude
μ₂	13.222222	term-magnitude
μ₃	111.222222	term-magnitude
μ₄	1247.296296	term-magnitude

Continuous — Exponential Distribution

Field	Value
Variable type	Continuous (CRV)
Lower bound	0
Upper bound	inf
PDF	2 * exp(-2*x) if x >= 0 else 0

Expected output:

Moment	Value	Method
μ₁	0.500000	gauss-laguerre+quad
μ₂	0.500000	gauss-laguerre+quad
μ₃	0.750000	gauss-laguerre+quad
μ₄	1.500000	gauss-laguerre+quad

Continuous — Normal Distribution (exact)

Field	Value
PDF	exp(-(x**2)/2) / sqrt(2*pi)
Bounds	-inf to inf

Expected output:

Moment	Value	Method
μ₁	0.000000	sympy-exact
μ₂	1.000000	sympy-exact
μ₃	0.000000	sympy-exact
μ₄	3.000000	sympy-exact

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Input Syntax

The expression field accepts natural mathematical notation. Common auto-corrections applied by the parser:

You type	Interpreted as
0.3 * 0.7^x	0.3 * 0.7**x
e^(-2*x)	exp(-2*x)
2x	2*x
|x|	abs(x)
ln(x)	log(x)
√(x)	sqrt(x)
x ≤ 1	x <= 1
λ	lam

See the Input Syntax Guide for the complete reference.

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Project Structure

ToFUL/
├── app.py              Streamlit frontend (UI, layout, plots)
├── core.py             Computation engine (moments, validation, convergence)
├── toful_parser.py     Input normalisation pipeline (two-pass)
├── requirements.txt    Runtime dependencies
└── docs/               Sphinx documentation source
    ├── conf.py
    ├── index.rst
    ├── getting-started/
    ├── user-guide/
    ├── examples/
    ├── theory/
    ├── api-reference/
    └── troubleshooting/

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Numerical Methods

Discrete series convergence cascade

1. Term magnitude — last 10 terms all below tolerance
2. Wynn ε-algorithm — recursive epsilon-table extrapolation
3. Aitken Δ² — three-point partial-sum extrapolation
4. Cohen-Villegas-Zagier — optimal for alternating series
5. Geometric ratio bound — tail estimate from asymptotic ratio

Continuous integration

Domain	Method
[0, ∞)	Gauss-Laguerre quadrature (64 nodes) + quad cross-check
(-∞, ∞)	Gauss-Hermite quadrature (64 nodes) + quad cross-check
Finite [a, b]	SciPy adaptive Gauss-Kronrod
Any, precision > 12dp	mpmath tanh-sinh (arbitrary precision)
No conditionals	SymPy symbolic integration (exact)

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Dependencies

Package	Purpose
streamlit	Web UI framework
numpy	Array operations, vectorised evaluation
scipy	Adaptive quadrature, Gauss nodes, special functions
sympy	Symbolic integration (optional but recommended)
mpmath	Arbitrary-precision arithmetic (optional)
plotly	Interactive distribution and moment plots
pandas	Results table and CSV export

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Documentation

Full documentation is hosted on ReadTheDocs: toful-rtd.readthedocs.io

To build locally:

cd docs
pip install sphinx sphinx-rtd-theme
make html
# open docs/_build/html/index.html

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Contributing

Contributions are welcome. See CONTRIBUTING for the development setup, code style guide, and pull request process.

When reporting a numerical bug, include the exact Range and PMF/PDF strings, the moment order, and the analytical expected value.

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License

MIT License — see LICENSE for details.

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Built with Streamlit · NumPy · SciPy · SymPy · mpmath · Plotly