A distribution of joint samples

Question

Let's assume we sample two distributions: $$x_1, x_2,\ldots,x_n \sim N\left(\mu_x, \sigma_x^2\right)$$ $$y_1, y_2,\ldots,y_m \sim N\left(\mu_y, \sigma_y^2\right)$$ If we consider those samples together, so we have a permutation of $x_1, x_2,\ldots,x_n, y_1, y_2,\ldots,y_m$ - can we determine the corresponding distribution?

Answer 1

I think there's a bit of confusion here -- you have samples, but they come from two distributions. What is the distribution you are after?

Assuming that you're interested in the distribution that a randomly selected sample from your $n+m$ samples would come from, it's a mixture model: Let $B$ be a Bernoulli variable with probability $n/(m+n)$. Then your distribution is $BX+(1-B)Y$.