Common Errors

This page lists the most frequent error messages in ToFUL, explains what causes each one, and gives concrete fixes.

Validation Errors 

“PMF sums to X ≠ 1.0”

What it means: ToFUL evaluated your probability mass function over the given support and the total came out to a value other than 1.

Common causes and fixes:

Wrong parameter value

# Bug — probabilities sum to 0.7, not 1.0
0.3 * (0.4 ** x) if x >= 0 else 0

# Fix — use the correct base: q = 1 - p
0.3 * (0.7 ** x) if x >= 0 else 0

Forgotten normalising constant

# Bug — unnormalised Poisson
(3 ** x) / factorial(x) if x >= 0 else 0

# Fix — include exp(-lambda)
(exp(-3) * 3**x) / factorial(x) if x >= 0 else 0

PMF defined only on a subset of the given range

If your range is 1,2,3,4,5 but the PMF returns 0 for some values, ensure probabilities over the non-zero values still sum to 1.

Finite range too narrow for an infinite-support distribution

If you entered a finite range like 0,1,2,3,4,5 for a Geometric distribution, only the probability in that window is summed — the tail is excluded. Use the infinite notation 0,1,2,3,... instead.

“Negative probabilities detected at x = […]”

What it means: The PMF returned a negative value at one or more support points.

Common causes:

Sign error in the formula — check for an extra minus sign.
Condition in the if guard is inverted, causing the negative branch to fire when the positive one was intended.

# Bug — returns negative for x > 0
-0.3 * (0.7 ** x) if x >= 0 else 0

# Fix
0.3 * (0.7 ** x) if x >= 0 else 0

“PDF integrates to X ≠ 1.0”

What it means: Numerical integration of your PDF over the given bounds did not return 1.0 within the precision tolerance.

Common causes and fixes:

Wrong bounds

# Bug — Exponential(2) integrated only to 5, missing most of the tail
Lower: 0   Upper: 5

# Fix
Lower: 0   Upper: inf

Missing guard

# Bug — evaluates e^x for x > 0, which grows without bound
PDF:  exp(-x)

# Fix — restrict to the correct support
PDF:  exp(-x)  if  x >= 0  else  0

Wrong normalising constant

Use the validation step to read off the actual integral, then divide your PDF by that value and recompute.

“Negative PDF value at x ≈ […]”

ToFUL spot-checks 100 test points before integrating. If any evaluate negative, validation stops immediately.

Ensure the PDF is non-negative everywhere on the support. If you are modelling a signed function for other purposes, ToFUL is not the right tool for that calculation.

Syntax Errors 

“Syntax error: invalid syntax (position N)”

The expression could not be parsed as valid Python after auto-correction.

Most common causes:

What was typed	Fix
`x^2`	Not an error — auto-corrected to `x**2`
`2^x`	Not an error — auto-corrected to `2**x`
`f(x) = exp(-x)`	Remove `f(x) =` — enter only the RHS
`exp[-x]`	Use round brackets: `exp(-x)`
`P(X=x) = 0.5`	Enter just the value: `0.5`
`x >= 0 and <= 1`	Write: `0 <= x <= 1`
`lambda * exp(-x)`	`lambda` is a Python keyword; use `lam`

Unmatched brackets

Count opening and closing brackets. Each ( must have a matching ) and vice versa.

# Bug — unmatched bracket
exp(-2*x if x >= 0 else 0

# Fix
exp(-2*x) if x >= 0 else 0

Range Errors 

“Cannot parse range ‘…’”

The range string could not be converted to a list of numbers.

Common causes:

Spaces without commas: 1 2 3 → use 1,2,3
Non-numeric characters: x=0,1,2 → use 0,1,2
Decimal comma instead of decimal point (locale issue): 1,5 as “one point five” → use 1.5

Fixes:

Reduce the maximum moment order.
Use the gamma function instead of factorial for large arguments.
Reformulate the expression to avoid intermediate large numbers:

# Bug — computes 100! internally, overflows
factorial(100) / (factorial(x) * factorial(100-x))

# Fix — use log-space or gamma
exp(log(gamma(101)) - log(gamma(x+1)) - log(gamma(101-x)))

“Computation error: ZeroDivisionError”

Division by zero in the PMF or PDF expression.

Check for a denominator that can be zero at any support point:

# Bug — divides by x, which is 0 for the first support value
1 / x  if  x > 0  else  0.5

# Fix — guard the denominator
1 / x  if  x > 0  else  0