I have two joint distributions say $f_1(x,y)$ and $f_2(x,y)$. Let $f(x,y)$ be the weighted sum of $f_1$ and $f_2$ such that

$$f=w_1\cdot f_1+w_2\cdot f_2;$$

How do make $100$ random draws from this new distribution $f$?

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    $\begingroup$ What are f, f1, and f2, exactly? Probability density functions? $\endgroup$ – Kodiologist Jan 8 '17 at 1:49
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    $\begingroup$ If $f_1$ and $f_2$ are known pdfs and $w_1$ and $w_2$ are known scalars, then in what sense is this new distribution "unspecified"? $\endgroup$ – JiK Jan 8 '17 at 13:26

Presumably $w_1+w_2=1$ and $w_1,w_2 \geq 0$, so $f$ is a convex combination of $f_1,f_2$ and therefore a valid distribution (a mixture of $f_1,f_2$).

Generate a Bernoulli($w_1$) random variable (i.e. flip a coin, heads = 1 with probability $w_1$). If it's $1$, draw a sample from $f_1$. If its $0$, draw a sample from $f_2$. You can use the law of total probability to see that this is indeed gives you a sample drawn from distribution $f$, by conditioning on the two values the Bernoulli r.v. takes and applying the law of total probability.

Do this 100 times to get 100 samples from $f$.

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    $\begingroup$ Of course we assume f_1 and f_2 are known distributions. Also the distributions are bivariate which affects how you sample from f_1 or f_2.. $\endgroup$ – Michael Chernick Jan 8 '17 at 3:30

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