Context of the Poisson distribution: $ n rightarrow infty $ and $ lambda: = np $ where $ p in (0,1) $ can be considered as a probability of success, $ n $ is the amount of possibilities and $ lambda $ as the average or expected value of success.

You can see the full proof of converting the probability mass function of a binomial distribution in Poisson here on page 3, although I don't think you need to read it for this question:

https://mbernste.github.io/files/notes/Poisson.pdf

A key step in this test is this relationship:

$$ e ^ {- lambda} = lim_ {n rightarrow infty} left (1- frac { lambda} {n} right) ^ n $$

I understand this equality and your test when $ n $ and $ lambda $ they are independent of each other, but have trouble accepting equality when there is a relationship, namely $ lambda = np $. Assuming $ p $ It is fixed and not zero, $ z: = – frac {n} { lambda} $and using the limit definition of $ e $, the proof of this equality is as follows:

$$ lim_ {n rightarrow infty} left (1 – frac { lambda} {n} right) ^ n = lim_ {n rightarrow infty} left (1 + frac {1} { left (- frac {n} { lambda} right)} right) ^ { left (- frac {n} { lambda} right) left (- lambda right)} = lim_ {z rightarrow infty} left (1+ frac {1} {z} right) ^ {z left (- lambda right)} = e ^ {- lambda} $$

I feel that this test is incomplete or has some errors, specifically:

- As $ z: = – frac {n} { lambda} $, shouldn't be the limit $ z rightarrow – infty $ instead? (Equality is still true)
- As $ n rightarrow infty $ and $ lambda: = np $should not $ lambda rightarrow infty $ so that $ e ^ {- lambda} = 0 $?