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I have read the answer here. Here the distinction is that

  • If $n\to\infty$ and $p\to0$ while $np$ approaches some positive number $\lambda,$ then the binomial distribution approaches a Poisson distribution with expected value $\lambda.$

  • If $n\to\infty$ as $p$ stays fixed, and $X\sim\operatorname{Binomial}(n,p)$ then the distribution of $(X-np)/\sqrt{np(1-p)}$ approaches the standard normal distribution, i.e. the normal distribution with expected value $0$ and standard deviation $1.$

I am finding it hard to wrap my head around this. In the derivation of the central limit theorem nowhere is $p$ taken into consideration. So even if $p$ is very small, according to CLT the standardized Binomial should limit to a standard normal. And the two limiting behaviors are both for $n \to \infty$ Please help me understand this concept a bit more. How can CLT not be valid when $p$ is really small?

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    $\begingroup$ On first pass, my instinct is that ambiguity arises here due to insufficient precision in how the semantics of "approaches" is interpreted mathematically. $\endgroup$ – microhaus Apr 10 at 12:17
  • $\begingroup$ The answer you reference clearly articulates the fundamental point, beginning with "It is sloppy to say something approaches something depending on n as n→∞, unless it is precisely defined and not meant literally...." $\endgroup$ – whuber Apr 10 at 12:25
  • $\begingroup$ You appear to be making a common mistake about the central limit theorem: stats.stackexchange.com/questions/473455/…. The central limit theorem is about a sampling distribution, not about the original population. $\endgroup$ – Dave Apr 10 at 12:35
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The difficulty disappears when you are careful in formulating the limits. In the first case, $p$ is not constant, so it would be more precise to write it as $p_n$, as $p$ varies with $n$. We can write $n \cdot p_n \to \lambda>0$ another way as $p_n \sim \lambda/n$, where $\sim$ means that the quotient between the two sides converges to unity with $n \to\infty$.

For the second case, $p$ is constant, and however small, when $n$ is large enough, $np$ is no longer small. The CLT is still valid when $p>0$ is small and constant.

To understand this better, you could try to use the CLT for IID variables in the first case. Write the binomial out as a sum of $n$ IID Bernoulli variables, as $X_n= B_1 + \dotsm + B_n$. Now check the assumptions of the CLT. You will find that the $B_1, B_2, \dotso, B_n$ all must have the same distribution, and that distribution should not depend on $n$. Is that the case?

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