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Conditions for approximating binomial distribution to poisson distribution

So, a little context. The image you see is from the GCE A-LEVEL syllabus where they have defined the conditions for approximating binomial to poisson. But why did they have mention that the approximation is only possible when n > 50 and np/mean < 5? What is special about these two conditions? Is there a graphical or a proof-based reason?

THANK YOU FOR THE ANSWER(S)

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    $\begingroup$ "Possible" and "appropriate" don't mean the same thing. $\endgroup$
    – whuber
    Commented Sep 22, 2022 at 11:31

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That should be $np\geq 5$ and $n(1-p) \geq 5$ for any approximation to be taken on Binomial Distribution. The approximation can be attributed to Central Limit Theorem.

This type of approximation is defended in another way also. Binomial distribution has PMF: $$ f(X = x) = nCx~ p^x (1-p)^{n-x} $$ with $x = 0, 1, \ldots, n$. Now when $n \rightarrow \infty$ and $p \rightarrow 0$ the computing the quantity $nCx$ becomes cumbersome and $p^x$ and $(1-p)^{n-x}$ also become unnecessarily cumbersome. Then assuming $np = \lambda$ it becomes the mean of Poisson Distribution and the quantity $np(1-p) \rightarrow np$ as $1-p \rightarrow 1$ i.e. variance of the Binomial Distribution becomes the Variance of Poisson Distribution.

This is how the Binomial Distribution is approximated as Poisson Distribution. Binomial Distribution can be approximated as a Normal Distribution too.

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    $\begingroup$ Although these points are correct, neither really explains why "50" and "5" appear in the rule of thumb quoted in the question. $\endgroup$
    – whuber
    Commented Sep 22, 2022 at 11:40
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    $\begingroup$ @whuber surely it comes from the same line of arguments that produced the fact that the sample mean is normal whenever $n\geq30$ ;) $\endgroup$ Commented Sep 22, 2022 at 13:40
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Going by Prof G E P Box's quote "in statistics no model is perfect but some are useful", modeling data using Probability Distributions also fits this quote very well, one can use any distribution anywhere if the constraints based on assumptions are relaxed - but the estimation error will be varying depending upon what approximation is being used. These numbers like $50$, $5$ they are decided based on simulated runs on real data. Earlier not even 10 years back, it used to be $30$ as the number of samples which was used for Normal distribution to be assumed for any distribution. Generally speaking Statistics always prefer "More the Better". But these are all sample based Statistical methodology if we enter into Big Data then almost all the concepts of Statistics starting from Hypothesis Testing, Confidence Interval Estimation etc severely fail.

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