Approximating the number of successes in independent trials: Binomial versus Poisson

Let I1,I2,.,In be independent Bernoulli random variables with P(Ii=1)=1.P(Ii=0)=pi, 1.i.n, and W=.ni=1Ii, .=EW=.ni=1pi. It~is well known that if~pi's are the same, then~W follows a~binomial distribution and if~pi's are small, then the distribution of~W, denoted by~LW, can be well approximated by the Poisson(.). Define r=..., the greatest integer~.., and set .=....., and~. be the least integer more than or equal to max{.2/(r.1.(1+.)2),n}. In this paper, we prove that, if r>1+(1+.)2, then d.<d.+1<d.+2<.<dTV(LW,Poisson(.)), where dTV denotes the total variation metric and dm=dTV(LW,\break\Bi(m,./m)), m... Hence, in modelling the distribution of the sum of Bernoulli trials, Binomial approximation is generally better than Poisson approximation.