The probability density of the total IBD length over a single autosome in unilineal relationships

Several authors have studied identity by descent (IBD) by way of a continuous recombination process along a chromosome. Despite its potential uses in, for example, gene mapping or delineation of biological relationships there has been no exact algebraic result given for the probability density function of the IBD proportion in any familial relationship. Other authors have derived algebraic approximations in the case of half-sibs by way of the Poisson clumping heuristic and used computational methods to compute the distribution function of the IBD sharing for unilineal relationships. Here we provide a general numerical method for finding the density of IBD sharing that could be applied to any unilineal relationship and more importantly we derive algebraically an expression for the density for a grandparent-grandchild relationship. Initially we assume that recombination events occur at random along a chromosome, then go on to show how the method could be extended to incorporate a form of genetic interference.