Normal approximation to Bernoulli variable 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?

The error should not add up. You get as many times the error positive as negative.
If you are summing the variables that you approximate then in the end you will effectively determine the variance and mean of the final distribution by means of algebraic rules for the mean and variance of adding correlated variables.
$$\mu_{X+Y} = \mu_{X} + \mu_Y$$
$$\text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X,Y)$$
and it will be as if you are doing a single normal approximation for the end result. So no errors add up.
