Galton-Watson process

The Galton-Watson process isstochastic process arising from Francis Galton's investigation ofextinctionsurnames.

Assume, as was taken quitegrantedGalton's time, that surnamespassed onall male children by their father. Supposenumber ofman's sonsberandom variable distributed onset { 0, 1, 2, 3, ...}. Further supposenumbersdifferent men's sonsbe independent random variables, all havingsame distribution.

Thensimplest substantial mathematical conclusionthat ifaverage number ofman's sons1 or less, then their surname will surely die out,if itmore than 1, then theremore than zero probability thatwill survive forever.

This isfairly stubby article. Add more on this subject if you can.