# Is there a numerical solution to a mixture model of two normal distributions?

I'm building a mixture model with the two normal distributions $\mathcal{N}(\mu_1,\sigma_{1}^{2})$ and $\mathcal{N}(\mu_2,\sigma_{2}^{2})$. So, the density function is $$f(x) = p_1 N(x; \mu_1, \sigma_1^2) + p_2 N(x; \mu_2, \sigma_2^2),$$ where $p_1+p_2=1$, and $$N(x;\mu,\sigma) = \frac{1}{\sqrt{2\pi \sigma^2}}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\}.$$.

Suppose I have all the sampling data, is there some numerical solution or formula that could derive $p_1$, $\mu_1$, $\sigma_1$ and $p_2$, $\mu_2$, $\sigma_2$?

• The standard approach to estimating the parameters of a mixture of normal distributions is to use an Expectation-Maximisation algorithm. – waferthin Aug 5 '13 at 11:41

If the data includes this indicator variable you might simply split the data in two sub-samples corresponding to the distribution from which the data originates, and fit the two normal distribution separately using maximum likelihood. The parameters $p_1$ and $p_2$ can be estimated by the proportion of samples that come respectively from the first and second normal distribution.