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If I have (for example) normal distributions $p_1(x)$ and $p_2(x)$, what PDF (if any) will describe 1000 samples from p1 and 500 samples from p2?

some PDFs

i.e. is there a PDF describing the green histogram $x_1+x_2$?

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  • $\begingroup$ $x_1+x_2$ is not the ideal way of describing combination of two samples. $\endgroup$
    – gunes
    May 30 '21 at 19:43
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If you want to assign priors to distributions, i.e. a RV is $2/3$ likely to be drawn from $p_1$ and $1/3$ likely to be drawn from $p_2$, then the new PDF will look like $$p(x)=\frac{2}{3}p_1(x)+\frac{1}{3}p_2(x)$$ And, its histogram will be very much alike the green curve above.

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  • $\begingroup$ @JarleTufto what I describe is a mixture distribution. $\endgroup$
    – gunes
    May 30 '21 at 19:42

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