I teach statistics in a community college. In my class I gave an example of image noise reduction in cosmology. For example, an object is exposed for time T and we got image $C_1(i,j)$. $C_1$ is the counts of pixel $(i,j)$ in the camera. With the lens of the camera covered, we got image $C_2(i,j)$ in a longer time of $a\times T$. Both $C_1$ and $C_2$ follow Poisson distribution. The result image after processing is $C=C_1-\frac{C_2}a$. Now my question is what the variance of $C$ should be?
I think it should be $var(C)=var(C_1)+\frac{var(C_2)}{a^2}$. But one of my students said since the count follows Poisson distribution, $var(\frac{C_2}a)$ should be equal to $\frac{C_2}a$, not $\frac{C_2}{a^2}$. It sounds weird to me, because if it is like that, $var(C)$ will be almost a constant no matter how long you take the measurements, and there will be no more benefit with long measurements. A scaled Poisson distribution is not Poisson anymore. But I cannot persuade him. Is my logic correct or what did I miss? Can anybody give some other example to explain this problem? Many thanks!
Thanks for your clear explanation! (@ whuber) I fully agree that the variance should be estimated by regular uncertainty propagation rules. However, I guess my student's idea is that since the process of noise generation follows Poisson distribution, intrinsically, when we get a mean of $\frac{a\nu}{a}$, we also get a variance of the same value due to the characteristics of the process. It is different from other cases such as measuring the weight of an apple. The uncertainty may depend on the smallest unit of the scale and how many times it was measured, independent of the weight of the apple. In Poisson, the mean and variance are linked and identical. But this conflicts with traditional uncertainty propagation rules and scaling of distribution. I don't know whether his view is correct or not.