Derviation of Haldane's mapping function

Key assumption: The number of crossovers is Possion distribution with mean m.

let m = expected number of crossover

Prob(observe crossover) = Prob (1 CO) + Pr(3 Co) + .. + Pr(2k+1 CO) + ...

Assume cross-overs follow a Poisson distribution so that

Pr(2k+1) = ( e^-m m^2k+1 )/ (2k+1)!

hence c = Pr(obs co) = Sum Pr(2k+1)

= e^-m Sum ( m^2k+1)/ (2k+1)! ) = (1 - e^-2m]) /2