Question

# Which of the following binomial distributions can be well approximated by a normal distribution? A Poisson distribution? Both? Neither? (d)n=1000,p=.001

Normal distributions
Which of the following binomial distributions can be well approximated by a normal distribution? A Poisson distribution? Both? Neither?
(d)$$n=1000$$,$$p=.001$$

2020-12-04
The Rule of five:
The normal with means np and np q can be used to approximate the binomial distribution with parameters n and p if $$npq > 5$$.
Here, $$n = 1,000$$,$$p = 0.001$$.
$$npq = 1,000 \times 0.001 \times 0.999 = 0.999 <5$$</span>
Normal approximation cannot be used for the binomial distribution with $$n = 1,000$$,$$p = 0.001$$.
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 npis large.
The value of np is obtained as shown below:
np = 1,000 \times 0.001=1
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 $$n = 1,000$$, $$p = 0.001$$ cannot be approximated to a normal distribution and it can be approximated to a Poisson distribution.