# Summing Bernoulli distributions with noise [duplicate]

EDITED

John is playing a game on $$n$$ days, each day being independent.

On each day $$i$$, his probability of success is $$p_i$$.

We have $$\frac{1}{n} \sum_{i=1}^n p_i = p$$, and typically, the standard deviation $$\sigma$$ of these $$p_i$$ is small. $$\sigma$$ is known.

So I have a succession of $$Bernoulli(p_i)$$. I want to model the probability of having k success after n trials, using $$p$$ and $$\sigma$$ for instance.

I can approximate this with a Binomial distribution, but even though it's close it still leads to biases. Any thoughts on how to improve this?

## marked as duplicate by Tim♦Jan 25 at 14:19

• Could you explain in what sense the sum of 0/1 variables can be interpreted as a "probability of success"? What shall we do about the fact that your Normal model for $\epsilon$ places some chance of $p$ lying outside the interval $[0,1],$ which is impossible? Regardless, is $\sigma^2$ known or not--and if not known, what data do you have to estimate it? This appears to be an effort to simplify some real problem, but the simplification seems to be creating confusion: what is the actual problem? – whuber Jan 24 at 23:19
• Thanks for your comments, I fixed the definition problem of $p$. Also, $\sigma$ is known. – MaximeKan Jan 25 at 3:55