Quick Start Guide
This guide walks you through calculating your first moments in ToFUL. We will compute the first four moments of a Geometric distribution — a classic example that exercises both infinite discrete support and the full moment pipeline.
Step 1 — Open the Calculator
Visit toful1.streamlit.app (or run
streamlit run app.py locally) and click Open Calculator.
Step 2 — Choose Variable Type
In the sidebar, the Variable Type selector defaults to
Discrete (DRV). Leave it there — the Geometric distribution is
discrete.
Step 3 — Enter the Support
In the Range field type:
0,1,2,3,...
The trailing ... signals an infinite series. ToFUL detects the
arithmetic pattern (step = 1) and extends the support automatically
up to the configured maximum terms (default 200).
Step 4 — Enter the PMF
In the Probability Function field type:
0.3 * (0.7 ** x) if x >= 0 else 0
This is the PMF of a Geometric distribution with success probability
\(p = 0.3\). The if x >= 0 else 0 guard ensures the function
returns 0 outside the support.
Tip
You can also type 0.3 * 0.7^x if x >= 0 else 0 using a caret for
exponentiation — ToFUL’s parser will auto-correct it to **.
Step 5 — Choose Moment Reference
Leave Moment Reference set to Origin (a = 0) to compute raw
moments.
Step 6 — Set Moment Order
Set Max order (r) to 4 to compute μ₁ through μ₄.
Step 7 — Compute
Click Compute Moments. ToFUL will:
Validate that the PMF sums to 1 over the support.
Compute each moment using the convergence-acceleration cascade.
Display results across five tabs.
Understanding the Output
Moments Tab
Four metric cards appear, one per moment order. Each card shows:
The moment symbol (μ₁, μ₂, μ₃, μ₄)
The computed value
The numerical method used (e.g.
term-magnitude,wynn-epsilon)
For a Geometric distribution with \(p = 0.3\):
Order |
Value |
Interpretation |
|---|---|---|
μ₁ |
2.3333… |
Mean = (1-p)/p = 7/3 |
μ₂ |
13.2222… |
Second raw moment E[X²] |
μ₃ |
111.2222… |
Third raw moment E[X³] |
μ₄ |
1247.2963… |
Fourth raw moment E[X⁴] |
Statistics Tab
Switch the Moment Reference to Mean (a = μ) and recompute to see
derived statistics: variance, standard deviation, skewness, and kurtosis.
Distribution Tab
Shows a stem plot of the PMF annotated with the mean, alongside a bar chart comparing the magnitude of each moment.
Table Tab
A downloadable CSV table with all moment values, methods, and convergence status.
Convergence Tab
Per-moment diagnostic entries showing which convergence method fired and the associated info string.
Next Steps
Discrete Random Variables — full guide to discrete random variables
Continuous Random Variables — computing moments of continuous distributions
Discrete Distribution Examples — more worked examples
Input Syntax Guide — everything you can type in the function field