I’m trying to figure out the best split of time between measuring either background or signal+background in a counting experiment with the following conditions in two separate cases:

1. Known signal and background mean count rates, $s$ and $b$.
2. No prior knowledge about the mean signal count rate.

Assuming I have only a limited time T to perform measurements and there is no timing uncertainty.

My first attempt was to assume the errors would be simply $\sqrt{N}$, and I should try to minimise the error in the $(s+b)$ rate minus the $b$ rate for a given $x \in [0,1]$ as a fraction of T spent measuring $(s+b)$, however, this gets me to:

$\Delta (s+b)_m = \sqrt{s \cdot T \cdot x}$ $\Delta b_m = \sqrt{b \cdot T \cdot (1-x)}$

Now, if you want to calculate: $\dot{s}_m=\frac{(s+b)_m}{T\cdot x}-\frac{b_m}{T\cdot (1-x)} $ $\Delta s_m = \sqrt{(\Delta (s+b)_m/T\cdot x)^2+ (\Delta b_m/ T\cdot (1-x))^2}$

This function isn’t minimisable in my case. Is the approach wrong?

I’m trying to figure out the best split of time between measuring either background or signal+background in a counting experiment in the case where we have no prior knowledge about the mean signal count rate.

Assuming we have only a limited time T to perform measurements and there is no timing uncertainty.