I am given a population $P$ that is equally divided into subsets $A$ and $B$. I know that a property $H$ of the population $P$ is normally distributed with mean $\mu_1$ and variance $\sigma_1^2$ for subset $A$ and $\mu_2$ and variance $\sigma_2^2$ for subset $B$. The task at hand is to find the exact median value of $H$ for the entire population $P$.
This is a Gaussian mixture with equal weights. I know that for a Gaussian distribution, the mean and the median are equal. I am trying to set up an equation based on the given data that is going to allow me to determine the median. I believe that the density of the Gaussian mixture is:
$$ f(h) = 0.5f_A(h) + 0.5f_B(h) $$
where $f_A$ is the Gaussian for subset $A$ and $f_B$ the Gaussian of subset $B$.Now the standard way to handle this problem for distributions of continuous random variables is to solve for $m$ the equation:
$$ \int_m^{\infty} f(h)dh = \frac{1}{2} $$
This does not look like a good strategy for this problem (for computational reasons -- although I may be totally off). If anyone could suggest a nicer, more elegant approach, I would deeply appreciate it!