We want to calcualte the distribution of the distance to the first crossover event on a long chromosome. We call this distance \(X\).
Let \(Y\) be the number of recombination events over a length of \(d\) centimorgan. This number is Poisson distirbuted with parameter \(\lambda = d\). The probability that no crossover occured in this stretch of DNA is then
\[ P(Y = 0) = e^{-d}. \]
Note that if \(Y = 0\), this means that the distance to the first recombination event must be larger than \(d\). It follows that
\[P(X > d) = P(Y = 0) = e^{-d}.\] The CDF of X is therfore \[F_X(d) = P(X \leq d) = 1- P(X > d) = 1- e^{-d}.\]
This is the CDF of an exponential distribution with rate 1, whcih is what we wanted to show.