At several resources I found the claim that for large $n$ the Poisson distribution with mean $n$ is approximately normal with mean and variance $n$. Quite a few of these resources quote the central limit theorem. However, when I try to find some quantitative description of what "approximately" means, I am not successful.
I am interested in the following question.
Let $X$ have Poisson distribution with mean $n$ and $Y$ have normal distribution with mean and variance $n$.
What is the bound (in terms of $n$, $a$, $b$) for $d=|\mathrm{P}(a<X<b)-\mathrm{P}(a<Y<b)|$ ?
If someone could give me an expression or point me to a resource where I can find one, it would help. I could only find general statements about convergence, but nothing about the speed of convergence (except some demonstrations with numerical examples).