I have 2 normally distributed random variable $H_0$ and $H_1$, which are combined to give the weighted distribution $H$ as follows:
$H_0 \sim N(\mu_0, \sigma_0)$
$H_1 \sim N(\mu_1, \sigma_1)$
$$f_H = p * f_1(x) + (1-p) * f_0(x),$$
where $H$ has pdf $f_H$ and $H_1$ and $H_0$ have pdfs $f_1$ and $f_0$ respectively. The mean value $\mu$ of the combined distribution $H$ is:
$\mu = p * \mu_1 + (1-p) * \mu_0$
Now, what is the standard deviation $\sigma$ of $H$?
Is there a simple formula for $\sigma$, in terms of $p, \mu_0, \sigma_0, \mu_1, \sigma_1$?