Proportion of genes survived in offspring conditional on inheritance of flanking markers

In mammalian genetics and perhaps in human genetics as well, it is an interesting question as to how many offspring are needed in order to have a desired chance of preserving part of the entire genome of an individual. A more practical and perhaps more important question is: given k children and DNA marker data on a particular region of interest, what proportion of one's genes has been actually passed on to his children? To answer this question, I define the concept of identity by descent proportion, or IBDP for short. The IBDP is defined to be the proportion of genetic material shared identical by descent by a group of relatives in a specified chromosomal region. I provide a novel approach to computing the mean and variance of IBDP for k (> or = 2) half-sibs based on marker data, thus providing a means to compute the mean and variance of proportion of genes survived. I first show that each chromosome in an offspring can be represented by a two-state Markov chain, with the time parameter being the map distance along the chromosome. On this basis, I will show that IBDP can be written as a stochastic integral and that the computation of the EIBDP can be reduced to evaluating an integral of some elementary functions. Numerical examples are provided to illustrate the calculation.