Approximation for Beta distribution

Question

I've heard that, when the parameters $\alpha$ and $\beta$ of the beta distribution approach infinity, the distribution becomes approximately normal. But I have only seen a proof of such fact for the case $\alpha = \beta$.

Is it true that beta distributions with large parameters are approximately normal even when the parameters are not equal? If so, how big have the parameters to be for the approximation to be good?

I am willing to use this (or any other approximation that simplifies the calculations) with values $\alpha \ge 100$ and $10^4\alpha \ge \beta \ge 10\alpha$.

Answer 1

If I poperly remember, if you have a beta distribution with large values of parameters $(\alpha,\beta)$, it almost equivalent to a normal distibution with parameter $$\frac{\alpha }{\alpha +\beta} \sqrt{\frac{\alpha \beta }{(\alpha +\beta )^2 (\alpha +\beta +1)}}$$ I think that this comes from Taylor expansions of the beta distribution probability density function.

Use a mean of $\frac\alpha{\alpha+\beta}$ and a variance of $\frac{\alpha\beta}{(\alpha+\beta)^{2} (1+\alpha+\beta)}$