If n is very large and p is very small but np is a reasonable size (np = , say) then the distribution of Sn

If n is very large and p is very small but np is a ‘reasonable size’ (np = λ, say) then the distribution of Sn may be approximated by the Poisson distribution with parameter λ, as follows. For fixed k ≥ 0, write p = λ/n and suppose that n is large to find that P(Sn = k) =



n k



pk(1 − p)n−k

∼

nk k!



λ

n

k 

1 −

λ

n

n 

1 −

λ

n



−k

∼

1 k!

λke−λ. (2.20)