How are the normal and Poisson distributions related through the binomial?

[azay]

From the central limit theorem the binomial distribution can be approximated by a normal distribution N(0,1). But the binomial distribution can also be approximated by a poisson distribition.

Does this mean there is a relationship between the normal distribution and the poisson distribution (especially as n->infinity in B(n,p))?

I'm confused about this. To me it sounds like the normal distribution and the poisson distribution are equal then. Or does the standardisation in the central limit theorem make the difference?

Thanks

 

[mathman]

The difference depends on how you take the limit. To get a normal from binomial, you keep p constant as n -> ∞. To get a Poisson from a binomial, you keep np constant as n -> ∞.