# Which of the following binomial distributions can be well approximated by a normal distribution? A Poisson distribution? Both? Neither? (с)n=500,p=.001

Question
Normal distributions
Which of the following binomial distributions can be well approximated by a normal distribution? A Poisson distribution? Both? Neither?
(с)$$n=500$$,$$p=.001$$

2021-02-12
The Rule of five:
The normal with means np and npq can be used to approximate the binomial distribution with parameters n and p if $$npq > 5$$.
Here, $$n = 500$$,$$p = 0.001$$.
$$npq = 500 \times 0.001 \times 0.999= 0.4995<5$$</span>
Normal approximation cannot be used for the binomial distribution with $$n = 500$$, $$p = 0.001$$.
The direct approximation of the binomial by Poisson says that the binomial distribution with parameters n and p has the same distribution as the Poisson with the parameter np when np is large.
The value of np is obtained as shown below:
$$np = 500 \times 0.001 =0.5$$
Since the value of n is large and p is small and the value of np is small, the binomial distribution can be approximated to Poisson.
That is, the binomial distribution with $$n = 500$$, $$p = 0.001$$ cannot be approximated to a normal distribution and it can be approximated to a Poisson distribution.

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