# real analysis: take the limit of \$ left (1 – frac { lambda} {n} right) ^ n \$ where \$ lambda \$ and \$ n \$ are related in the context of the Poisson distribution

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:

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