statistical error on difference f(x) - x

Question

Suppose I've a random variable $X$ and a sample of it with size $N$. I count how many element of the sample fall in a specific range $a<x<b$ and I found $n$ entries, so if I use Poisson statistics I can say that the error on $n$ is $\sqrt{n}$.

Now suppose I've a function $f$, quite close to the unit function, and I define a new random variable $X'=f(X)$. Now the number of entries in $a<x<b$ is $n'=n+\delta n$.

I don't have an analitical from of $f$. I only now $n$ and $n'$.

What is the error on $n'-n$? How to take into account the correlation between them?

More difficult: suppose $f$ depends on a parameter $\lambda$ with error $\delta\lambda$.

Answer 1

Why would you use Poisson? If you have $n$ independent observations out of $N$ falling in $(a,b)$ with constant chance for each, wouldn't that be binomial rather than Poisson?

You can think of transforming not $X$ but (assuming invertibility, with $g = f^{-1}$), transforming $(a,b)$; the probability relates to the cdf of $X$ ($F(a') = F(g(a))$ and so on). The standard error of the binomial is $Np(1-p)$ where $p=F(b)-F(a)$ and $p' = F(b')-F(a')$ where the transformation of the endpoints works as above.