To preface, I'm working with some models in Python, and am making variations to test them against experimental data.
I have some model, which is a function $f(t,x,y)$. It depicts the rate at which an event occurs and has two parameters for other aspects of the model. I can find the number of events $N$ by integrating $f$ for fixed $x$ and $y$:
$$\text{Fiducial Model: }\int f(t, x_1, y_1)dt = N_1$$
Now I adjust my model parameters to $x_2$ and $y_2$, which gives me a different value for the number of events:
$$\text{Model Variation: }\int f(t, x_2, y_2)dt = N_2$$
I want to preserve the number of events between models, for all variations of $x_i$ and $y_i$ (within the bounds of my parameters, which are definite and known). To do this, I believe I should find some constant $A$ such that the second integral gives me the same number of events over the same integrated interval, i.e.
$$A\int f(t, x_2, y_2)dt = N_1$$
So would that simply mean that $$A = \frac{N_1}{N_2} = \frac{\int\text{Fiducial}}{\int\text{Varied}}\text{ ?}$$
In which case I just need to save the value $N_1$ (which is my fiducial model anyways), and multiply the integral by the ratio of the two values, right? My main concern was whether pre-computing my fiducial model is the best way to go about this, but mathematically it seems to be the case.
Edit: $dt$ not $dx$