When can we Central Limit Theorem approximation with good approximation?

I think we an use it when n(no. of trials) is large. But my textbook used this approx. by stating that since the expectation is large, we use the approx. I'm unable to understand this, would appreciate if someone helps.

Question

Solution Also since n is only 1, wouldn't this be a horrible approximation?

Poisson random number distribution looks like this. (from Wikipedia)

where $$E(X) = \lambda$$
It is easy to imagine what the curve will look like when $$\lambda = 1000$$.

The Poisson random number is the counting number of events within a specific time window.

And the time intervals between the events follow an exponential distribution.

If you generate one possible outcome (just like a simulation) how many exponential random numbers do we have to generate, for one possible outcome? On average 1,000 times.

If you are looking for a large $$N$$ this is the large $$N$$. it will not be always 1,000. but should be large enough to approximate Poisson as Normal.

You will observe one outcome, but one outcome is a combination of about 1,000 trials.

When we use 1,000 samples for mean/average, we count samples for $$N$$, we don't count how many sample means we got. We got only one sample mean,$$\bar X$$. Do we have $$N$$ = 1 or 1,000 ?

The calls are generated because you have perhaps $$10{,}000$$ customers attached to the switching office, who on average produce $$1000$$ calls in half an hour. Each customer can be considered as "a trial" (they either place a call, or do not), so you indeed have a very large number of "trials". If the expected number of calls had been every small--for example $$2$$--then it might not be so reasonable to assume that you are aggregating the behaviors of a large number of people.

Or you can simply rely on the observation that a Poisson distribution with a large expected value is well approximated by a normal distribution. This is a particular property of a particular distribution and does not require us to invoke the CLT, which is a much more general fact about general distributions.