Discrete Random Variables

A discrete random variable (DRV) takes values from a countable set — either finite (like a die roll) or infinite (like the number of coin flips before the first head). ToFUL computes moments for both cases.

Defining a Discrete Distribution 

A discrete distribution is defined by two inputs:

Support — the set of possible values \(\{x_1, x_2, \ldots\}\)
PMF — the probability mass function \(P(X = x)\)

The PMF must satisfy:

\[P(X = x) \geq 0 \quad \text{for all } x\]

\[\sum_{x} P(X = x) = 1\]

ToFUL validates both conditions automatically before computing anything.

Finite Support 

Enter a comma-separated list of values in the Range field:

1,2,3,4,5,6

Then enter a PMF that sums to 1 over those values:

# Fair six-sided die
1/6

# Non-uniform discrete distribution
0.1 if x == 1 else 0.2 if x == 2 else 0.3 if x == 3 else 0.2 if x == 4 else 0.1 if x == 5 else 0.1

For finite support, ToFUL evaluates the PMF at each value exactly and sums directly — no approximation involved.

Infinite Support 

End the range input with ... to signal an infinite series:

0,1,2,3,...

ToFUL detects the sequence pattern (arithmetic or geometric) and generates values up to the maximum terms setting (default 200, adjustable to 500 in Advanced).

Pattern detection examples:

Range input	Pattern detected	Extended as
`0,1,2,3,...`	Arithmetic (d=1)	0, 1, 2, 3, 4, …
`1,3,5,7,...`	Arithmetic (d=2)	1, 3, 5, 7, 9, …
`1,2,4,8,...`	Geometric (r=2)	1, 2, 4, 8, 16, …
`1,3,9,27,...`	Geometric (r=3)	1, 3, 9, 27, 81, …

Note

For distributions whose PMF decays quickly (e.g. Geometric with large \(p\), Poisson with small \(\lambda\)), 200 terms captures effectively all probability mass and the results are highly accurate. For heavy-tailed distributions, increase Max series terms in Advanced settings.

Convergence Acceleration for Infinite Series 

When computing moments over infinite support, ToFUL does not simply sum the first N terms and stop. It applies a cascade of convergence accelerators in order:

Term magnitude — if the last 10 terms are all smaller than the tolerance, the partial sum is accepted as converged.
Wynn ε-algorithm — extracts a limit estimate from the full sequence of partial sums via a recursive epsilon-table. Effective for a wide class of converging series.
Aitken Δ² — a lightweight three-point extrapolation, cross-checked against the current partial sum.
Cohen-Villegas-Zagier — optimal for alternating series; achieves near-machine-precision in exactly N evaluations.
Geometric ratio-bound — estimates the tail as a geometric series and adds it to the partial sum.

The Convergence tab shows which method fired for each moment order.

PMF Validation 

Before moments are computed, ToFUL validates the PMF:

Evaluates the function at every support value using vectorised NumPy.
Checks that no probability is negative.
Applies the convergence cascade to estimate the total probability mass.
Reports the sum and the convergence method used.

If validation fails, a detailed error is shown including the computed sum and the deviation from 1.

Common Discrete Distributions 

The following table shows how to enter standard distributions.

Distribution	Range	PMF expression
Bernoulli(p=0.4)	`0,1`	`0.4 if x == 1 else 0.6`
Binomial(n=5, p=0.3)	`0,1,2,3,4,5`	`(factorial(5)/(factorial(x)factorial(5-x))) 0.3*x 0.7**(5-x)`
Geometric(p=0.3)	`0,1,2,3,...`	`0.3 * (0.7 ** x) if x >= 0 else 0`
Geometric(p=0.3) †	`1,2,3,4,...`	`0.3 * (0.7 ** (x-1)) if x >= 1 else 0`
Poisson(λ=3)	`0,1,2,3,...`	`(exp(-3) * 3**x) / factorial(x) if x >= 0 else 0`
Negative Binomial(r=2, p=0.4)	`0,1,2,3,...`	`(factorial(x+1)/(factorial(x)factorial(1))) 0.4*2 0.6**x if x >= 0 else 0`

† The Geometric distribution has two common parameterisations — number of failures before the first success (support starting at 0) and number of trials until the first success (support starting at 1).

Tips for Discrete Distributions 

Use factorial(n) only for integer arguments. For non-integer generalisation, use gamma(n+1).
The PMF guard if x >= 0 else 0 is good practice — it prevents stray negative evaluations from contaminating the sum.
For Binomial distributions, compute factorial of small integers only. For large \(n\), reformulate using gamma.
Increase Calc precision to 10–12 for distributions with very small probabilities (e.g. Poisson with small \(\lambda\) and large support).