ToFUL — Tool for Uncertainty Learning
ToFUL is an interactive web application and Python computation engine for computing statistical moments of probability distributions — both discrete and continuous — with rigorous numerical methods, convergence acceleration, and real-time validation.
E[(X − a)ʳ] ← the r-th moment of X about point a
Whether you are a student exploring probability theory, a researcher verifying distributional properties, or a data scientist needing quick moment calculations, ToFUL provides accurate results with transparent methodology.
Try it now: toful1.streamlit.app
Getting Started
User Guide
Theory & Background
Troubleshooting
- Common Errors
- Debugging Functions
- Step 1 — Read the Auto-Correction Notice
- Step 2 — Check the Validation Badge
- Step 3 — Test a Single Value Mentally
- Step 4 — Verify the Guard Condition
- Step 5 — Check for Python Keyword Conflicts
- Step 6 — Isolate Multi-Part Expressions
- Step 7 — Check for Operator Precedence Issues
- Step 8 — Verify with a Known Distribution
- Diagnostic Checklist
- See also
- Convergence Issues
- Understanding the “Convergence Uncertain” Flag
- Fix 1 — Increase Max Series Terms
- Fix 2 — Increase Calc Precision
- Fix 3 — Check Whether the Moment Exists
- Fix 4 — Reformulate the PMF/PDF
- Fix 5 — Use a Tighter Range
- Fix 6 — Switch to mpmath Mode
- Diagnosing Slowly Converging Series
- Alternating Series
- Quick Reference — Which Fix to Try First
- See also
Development