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

Chesley 2021-02-11 Answered
Which of the following binomial distributions can be well approximated by a normal distribution? A Poisson distribution? Both? Neither?
(с)$n=500$,$p=.001$
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## Expert Answer

d2saint0
Answered 2021-02-12 Author has 89 answers

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, $n=500$,$p=0.001$.
$npq=500×0.001×0.999=0.4995<5$
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:
$np=500×0.001=0.5$
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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