Setup
Let X follow a fininte mixture model with density \begin{equation} f=\lambda f_1+(1-\lambda) f_2 \end{equation} Where $f_1$ and $f_2$ are both log-normal densities with parameters $(\mu_1, \sigma_1)$, and $(\mu_2, \sigma_2)$ respectively and $\lambda \in [0,1]$ is the mixing probability. Let $Z_i$ be a random variable with density $f_i$, i=1,2
I want to generate a set of N realisations from the random variable X.
In practice I saw two ways to do this:
Method 1
For each draw: choose with probability $\lambda$ a realisation from $Z_1$ and with probability $(1-\lambda)$ a realisation from $Z_2$.
Method 2
Draw N realisations from $Z_1$ and N realisations from $Z_2$, to obtain the two vectors $\mathbf{z}_1=(z_{11},z_{21}, ...z_{N1})^T $ and $\mathbf{z}_2=(z_{12},z_{22}, ...z_{N2})^T$.
Then obtain the desired vector of N realisations as
\begin{equation} \mathbf{x}=\lambda \mathbf{z_1}+(1-\lambda) \mathbf{z_2} \end{equation}
Question
I find Method 1 intuitive, but Method 2 seems strange. If we would impose limited support for $f_1$ and $f_2$ and allow, for example, only integer values, then the draws of X would contain values which are not in the support of either $Z_1$ or $Z_2$. This said, I did some simulations, and the histograms I plotted are identical. Hence, the question:
Are Method 1 and Method 2 asymptotically equivalent? Are they equivalent in small sample as well? Is one of the two methods just wrong?
Thanks a lot for your help!