Galton watson process
Aims and Objectives of Educational Psychologyġ.A Cognitive-Developmental View of Educational Psychology.
Introduction to Educational Psychology 2. For graphs with cycles, we can still use Galton-Watson processes which dominate their percolation, to derive useful results.After reading this article you will learn about:- 1. In percolation, for instance, any percolation on a tree is a Galton-Watson process, and can be analysed accordingly. These processes are also used in other fields in Probability. Note, however, that as the model allows for unbounded growth of the population, it might not be realistic in many situations. First, we can use it to describe populations ( biological and other) for more characteristics than just their surname. Today, Galton-Watson processes are employed as a basic probabilistic model in many fields. The case μ(D)=1 is the most interesting, of course. If μ(D) = 1, then either all Z's are (almost always) constantly 1, in which case there will almost always be precisely 1 Smith in every generation, and the Smith surname will live on forever, or Z's are not constants, in which case almost always the Smith surname will die out.with probability 1) the Smith surname will die out. If μ(D) > 1, then with positive probability the Smith surname shall live on (for ever).The expected number of Smiths at generation k is μ(D) k.Galton and Watson showed the following: Let μ(D)= EZ 1 (= EZ n, for all n) be the expected number of sons of any Smith. If, on the other hand, we allow for the possibility of having 0 sons, then there is always a positive probability of extinction: ur-Smith himself could have 0 sons, or even (with very low probability) once there are 1000000 Smiths, all will have 0 sons. Obviously, if P(Z 1=0)=0 (the probability of having no sons is 0) then this will never happen. The Smith family name will become extinct if at some generation there are no further sons. ur-Smith's lineage in this manner, and satisfy the above 2 properties. Obviously, despite general nastiness in computing the indices, we can construct Mr. The Z 2 sons of son #1 will have Z Z 1+2.ur-Smith shall have Z 2 sons son #2 shall have Z 3. Procure an infinite supply of IID random variables Z 1, Z 2.All Z xs are independent random variables.Z x has a distribution D, which is fixed for all males.Watson and Galton assumed this model for the offspring of a given male x: Let the random variable Z x be the number of sons of x. ur-Smith, its sons are his sons, and so on. Note that this is a random tree: the root is Mr. We need to trace the number of male descendents of a given male individual ("ur-Smith"?) to determine what happens to his surname. Would all surnames eventually become extinct? Assuming new surnames are invented (" Sting" might become one) or imported (" Singh") (or both (" Scolnicov")), it is conceivable that every surname would eventually become extinct, without the human race ( synonymous, of course, with the English) becoming extinct.Īssuming traditional usage, in propagating a surname only male scions count (the women, good Victorians that they undoubtably are, will either adopt their husband's surname and have children or keep their maiden name and remain childless note, however, that we may easily adapt the parameters of the model to suit other more modern circumstances). He knew that some surnames were no longer borne by anyone, others ("Smythe") were somewhat scarce, and still others ("Smith") were hugely popular. Galton asked the question if every surname would eventually become extinct. Despite its classical structure and the elementary derivations of its basic theorems, it, along with Markov processes, is one of the fundamental models of modern probability theory.
The same model is also commonly known as a " branching process". Henry William Watson (who solved it) in their study of the extinction of English surnames. The Galton- Watson process probabilistic model was developed by Sir Francis Galton (who proposed the problem) and the Rev.