# How can I efficiently approximate the sum of Bernoulli random variables for any number of summands in partial sum?

I have almost the same question as: How can I efficiently model the sum of Bernoulli random variables? But:

(1) The number of random variables for summation is ~ N=20 (case 1) or N=90 (case 2).

(2) $p_i$ ~ 0.13 (case 1)

(3) The precision of the model based on Poisson law is not enough.

(4) We need that our approx would be the good enough to model partial sums like these as well: $\sum_{i=k,N}{X_i}$, ( $k=1,N$ )

(5) We have empirical data for every $X_i$. The diagram shows that there is almost linear dependence for $Pr(X_i=1)$ for i=1,6 and then almost constant function or some small linear dependence (for i=7,20 ).

Really I'm not sure in (3) since the Le Cam's inequality looks like very general...

What class of model could we try?

• Sums of equally distributed bernoullis are binomial. They are slightly different in our case. We use $$p_i=p(1+i/N {\delta})$$ model now. But it should be some better model. Apr 19, 2011 at 8:57