On the genotype frequencies and generating function for frequencies in a dyploid model

Abstract

For a locus with two alleles ($I^A$ and $I^B$), the frequencies of the alleles are represented by $$ p=f(I^A)= \frac {2N_{AA} +N_{AB}} {2N} , \ q=f(I^B)= \frac {2N_{BB} +N_{AB}}{2N}$$ where $N_{AA},~N_{AB}$ and $N_{BB}$ are the numbers of $I^A I^A ,~I^A I^B$ and $I^B I^B$ respectively and $N$ is the total number of populations. The frequencies of the genotypes expected are calculated by using $p^2 ,~2pq$ and $q^2$. So in this paper, we consider the method of whether some genotypes is in Hardy-Weinburg equilibrium. Also we calculate the probability generating function for the offspring number of genotype produced by a mating of the $i$th male and $j$th female under a diploid model of $N$ population with $N_1$ males and $N_2$ females. Finally, we have conditional joint probability generating function of genotype frequencies.