Question

The closer p is to 0 or 1, the larger n must be in order for the distribution of p to be approximately normal. Please give me a more complete explanation than what is currently on Chegg. Thank you.

Answer 1

I assume, the question is when the Binomial Distribution tends to normal distribution

When p is the success probability and n is the number of trials, the Binomial Distribution is represented as

With mean = np and

sd =

When n becomes sufficiently large and p is small, the np becomes finite and the above distribution will follow Poisson with parameter

The next step is the normal approximation of Poisson Distribution

From the moment generating function of Poission distribution we find

and

both of which become negligibly small when is very large number. This suggests that Poisson distributin can be approximated by a normal distribution

So we can have this emperical relationship

Bi -> Po -> Normal; when p is small and n is sufficiently large