Calculator Inputs
The page stays single-column, while the calculator inputs adapt to large, medium, and small screens.
Example Data Table
Use this sample set to test the calculator quickly.
| Initial Ancestors | Generations | Trials | P(X=0) | P(X=1) | P(X=2) | P(X=3) |
|---|---|---|---|---|---|---|
| 2 | 8 | 3000 | 0.30 | 0.30 | 0.25 | 0.15 |
| 3 | 10 | 5000 | 0.20 | 0.25 | 0.30 | 0.25 |
Formula Used
Probability generating function: G(s) = Σ pksk
Mean offspring: m = Σ k pk
Offspring variance: σ² = Σ k²pk − m²
Expected size at generation n: E[Zn] = Z0 mn
Extinction by generation n for one ancestor: qn = G applied n times to 0
Extinction by generation n for Z0 ancestors: qnZ0
Eventual extinction: q is the smallest solution of G(q) = q
For one starting ancestor, the generation variance is:
If m ≠ 1: Var(Zn) = σ² mn−1(mn − 1) / (m − 1)
If m = 1: Var(Zn) = nσ²
For multiple initial ancestors, the calculator multiplies these one-ancestor results by Z0.
How to Use This Calculator
- Enter the number of initial ancestors, generations, simulation trials, and a safe population cap.
- Set the maximum offspring count K that your distribution will include.
- Fill the offspring probabilities for every count from 0 to K.
- Leave normalization checked if your probability entries do not sum exactly to one.
- Submit the form to see extinction measures, expected growth, simulated paths, and the generation table.
- Use the CSV and PDF buttons to export the computed table and summary.
FAQs
1. What does this calculator estimate?
It estimates extinction probabilities, expected population size, generation variance, survival chance, and simulation-based path summaries for a Galton-Watson branching process with a custom offspring distribution.
2. What is a Galton-Watson process?
It is a branching model where each individual independently produces a random number of offspring using the same probability distribution in every generation.
3. Why is the mean offspring number important?
The mean offspring value determines whether the process is subcritical, critical, or supercritical. That classification strongly affects long-run growth and extinction behavior.
4. Why can a supercritical process still go extinct?
Even when the average offspring count exceeds one, early unlucky generations can eliminate every lineage. The extinction probability falls below one, but it does not become zero automatically.
5. What does extinction by generation mean?
It is the probability that the population has already reached zero on or before a specific generation. Once extinction happens, later generations remain zero.
6. Why include Monte Carlo trials?
Simulation shows how random sample paths compare with theoretical expectations. It helps you see practical variation, extinction frequency, and potential divergence in finite experiments.
7. What does the population cap do?
The cap prevents extremely large simulated populations from consuming too many resources. If many trials hit the cap, raise it for a less truncated simulation.
8. Can I use probabilities that do not sum to one?
Yes. Keep normalization enabled and the calculator rescales the entries automatically. Disable it only when you want strict validation of a properly normalized distribution.