Poisson Approximation

IP(Xi = Yi ). (4.6)

The following proposition is a simple consequence of (4.4) and (4.6).

Proposition 4.3 If independent r.v.s Xi ∈ B( pi ) and Yi ∈ Π( pi ), pi ∈ [0; 1], then

( n∑

i=1 Xi ;

Yi

) ≤

pi (1 − e−pi ). (4.7)

The accuracy of the Poisson approximation for ∑n

i=1 Xi can be better if we choose Yi ∈ Π(λi ), where λi = − ln(1 − pi ).