Hi everyone just a technical question this time, (seriously at this point i don't think i could live without this site you are all life savers)
From A first course in probability: Section 4.7: The Poisson Random Variable Page:143-144
"The Poisson random variable has a tremendous range of applications in diverse areas because it may be used as an approximation for a binomial random variable with parameters $(n, p)$ when $n$ is large and $p$ is small enough so that $np$ is of moderate"
they then go to give a proof
Suppose $X\sim B(n,p)$ and let $\lambda=np$ $$P\{X=i\}=\frac{n!}{(n-i)!i!}p^{i}(1-p)^{n-i}$$ $$P\{X=i\}=\frac{n!}{(n-i)!i!}\left(\frac{\lambda}{n}\right)^{i}\left(1-\frac{\lambda}{n}\right)^{n-i}$$ $$=\frac{n(n-1)\cdots(n-i+1)}{n^{i}}\frac{\lambda^{i}}{i!}\frac{(1-\lambda/n)^{n}}{(1-\lambda/n)^{i}}$$ then for $n$ large and $\lambda$ moderate $$\left(1-\frac \lambda n \right)^n \approx e^{-\lambda}, ~\frac{n(n-1)\cdots(n-i+1)}{n^{i}} \approx 1, ~ \left(1-\frac{\lambda}{n}\right)^i \approx 1$$ hence for $n$ large and $\lambda$ moderate $$P\{X=i\} \approx e^{-\lambda}\frac{\lambda^{i}}{i!}$$
so my lack of understanding is the moderation of $\lambda$, in this instance we're choosing the number of trials to be large so i would assume that p must be relatively low in order to "modereate" $np$ but i have no idea what this moderation is in relation to.
how do we know when $np$ is moderate? for what boundries must np satisfy to be considered moderate? whats the definition of moderate in this regard.
is the idea of NP being moderate simple and i'm overthinking it? or overlooking something in the proof that should give a clarification to my issue?
thank you in advanced for the answers/comments.
To say that $\lambda$ is "moderate" actually means that as $n$ approaches $\infty$ and $p$ approaches $0,$ their product $np$ approaches $\lambda$ which is neither $0$ nor $\infty$ but somewhere between those two.
The argument you present assumes $np$ remains equal to $\lambda$ as $n\to\infty$ and $p\to0.$
"Moderate" might also be taken to mean that if $\lambda$ is either too big ("too close to $\infty$") or too close to $0,$ then the convergence as $n\to\infty$ and $p\to 0$ is slow.
An example with $\lambda = 3.4:$ \begin{align} \frac{100\times99\times98\times97}{4\times3\times2\times1} \left( \frac{3.4}{100} \right)^4 \left( 1 - \frac{3.4}{100}\right)^{100-4} & \approx 0.18929606 \\[15pt] \frac{1000\times999\times998\times997}{2\times3\times2\times1} \left( \frac{3.4}{1000} \right)^4 \left( 1- \frac{3.4}{1000} \right)^{1000-4} & \approx 0.18616366 \\[15pt] \frac{3.4^4 e^{-3.4}}{4\times3\times2\times1} & \approx 0.18582459 \end{align}