kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

Notes1. Our formulas obviously presuppose an absolute absence of any selection whatsoever.The biotype is reproduced under conditions of panmixia.2. See my paper (1922).3. By a pure race we designate a class which, when the crossing is interior, only producesindividuals of its own class.4. The case of two classes is obviously exhausted by the formulasf = ( + ), f 1 = ( + ) and f = p( + ) 2 , f 1 = q( + ) 2 .5. An essential part in our derivations, which were meant for biological applications, wasplayed by the restriction imposed on the signs of the coefficients. If we assume that theirsigns can be arbitrary, the solutions, except for formulas (16), can be of two types. The firstone corresponds to a linear function F that depends on five parameters. The second type ischaracterized by a quadratic function F and is represented by the formulas depending on fourparameters P, Q, d and d 1 :f = (1/4P)[P – Q + (d – 1) (d 1 – 1)S] [P – Q + (d + 1) (d 1 + 1)S], (18)f 1 = (1/4Q)[P – Q + (d – 1) (d 1 + 1)S] [P – Q + (d + 1)(d 1 – 1)S], = S 2 – f – f 1 .6.{Bernstein only explains the composition of the sum of the terms ih .}7. If n is the number of the pure races, then the number of all the classes is N = n (n + 1)/2.8. Because of the assumption that the crossing of races A 11 and A hh only produces only therace A 1h .9. Incidentally, the law of heredity expressed by equations (19) and representing a simplegeneralization of the Mendelian law, was applied when studying Aquilegia as investigated byBauer (Johannsen 1926, p. 581).{In this sentence, Bernstein made a grammatical error and“investigated by Bauer” is only my conjecture.}10. Our theoretical conclusions are fully confirmed by Morgan’s (1919) experimentalinvestigations.11. Otherwise we directly apply the theorem proved for n = 3 and obtain i = i ( 1 + 2 + ) 2 , i = 1, 2, 3.Hence we immediately prove the theorem also for n = 4.12. We consider the contrary case below.13.{In §3, see formula (8), Bernstein made use of the same letter S in another sense.}14.{It is difficult to understand the end of this sentence. The author wrote: In any case,after equating both parts of (29) to each other …}15. When 1 + 2 + … + n–1 + = 1.16. For the sake of definiteness we assumed that such terms are three in number, but ourreasoning will not change had we chosen another number.17. The case in which, apart from the hybrids, individuals of the parent classes can also beproduced, would have led to an appropriate generalization of the formulas.18. If the parents themselves are hybrids, a certain part of the offspring, depending on thekind {the sex? – difficult to understand the Russian text}of the parents, belongs to pure races,but the proportion of the different types of the hybrid offspring persists.