I'm looking for a normal approximation for a Bernoulli variable (so I can later sum multiple correlated approximated variables)
The trivial approximation is taking the mean and variance of the Bernoulli variable, and use those as parameters for the normal variable.
$$X \sim Bernoulli(p)$$ $$E[X]=p=\mu, Var[X]=p(1-p)=\sigma^2$$ $$\tilde{X} \sim N(\mu,\sigma^2)$$
This approximation has a built-in error (as analyzed here), which might add up when summing the correlated variables.
Is there a better choice for the normal variable parameters that will reduce the error of the approximation?