I have to show that $$ \frac{\binom{i}{N}^2}{\binom{2N}{N}}\left(\frac{i}{N}\right)^2= \frac{\binom{i-1}{N}^2}{\binom{2N}{N}}\left(1-\frac{i-1}{N}\right)^2 $$
My calculations:
$\frac{\binom{i}{N}^2}{\binom{2N}{N}}\left(\frac{i}{N}\right)^2 = \frac{\left( \frac{N!}{(N-i)!i!}\right)^2}{\left( \frac{(2N)!}{(2N-N)!N!}\right)} \left( \frac{i}{N}\right)^2 = \frac{(N!)^2}{[(N-i)!]^2(i!)^2} \frac{(N!)^2}{(2N)!} \frac{i^2}{N^2} $
further calculations did not lead to the result..