# Probabilities over the Union of Differently Distributed Populations

I'm working with a problem where the sample space is the union of two populations, each normally distributed. Specifically, I'm given that the heights of women are normally distributed, and the heights of men follow a (different) normal distribution, and am asked to work with the population as a whole (assuming equal numbers of men and women).

So, my first instinct is to say the pdf is $\frac 1 2$ the sum of the pdfs for men and women. However, this doesn't lend itself nicely to explicit computation. Is there a nicer way to deal with this problem (like writing it as a single normal distribution?) In particular, is there some trick to calculating the median here?

• The median is trivial if the standard deviations are equal, and difficult (i.e. requiring numerical methods or other approximations) if they are not. – Henry Oct 8 '14 at 21:36

The sum of independent normally distributed random variables is a normally distributed random variable, with a mean of the sum of their means and a variance of the sum of their variances.

$X\sim N(\mu_X, \sigma^2_X), Y\sim N(\mu_Y, \sigma_Y^2) \implies \frac{X+Y}{2}\sim N(\frac{\mu_X+\mu_Y}2, \frac{\sigma_X^2+\sigma_Y^2}{4})$

• Is this an accurate way to model this problem, though? I'm not looking at the average height of a randomly chosen man and woman. – Empiromancer Oct 8 '14 at 21:03