What I am keen to know is the relationship between $x_1, y_1$ and $x_2, y_2$ to keep the equality below correct knowing that $N_1 < N_2$ (for the sake of simplicity we can assume $N_2 = a\times N_1$ where $a \in \{2, 3, 4, ...\}$):
$$\int_{\frac{N_1}{x_1+y_1}}^{\frac{N_1}{x_1}} t^{N_1}e^{-t}dt = \int_{\frac{N_2}{x_2+y_2}}^{\frac{N_2}{x_2}} t^{N_2}e^{-t}dt$$
It is obvious that $\frac{x_2}{y_2}$ should be greater than $\frac{x_1}{y_1}$ to keep the equality correct, but I need more details. Thanks.