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We know that the weighted sum of CDF $$ F(x) = w_1 F_1(x) + w_2 F_2(x), \,\, w_1 + w_2 = 1 $$ is the CDF of the mixture distribution.

Is there a probabilistic interpretation for weighted sum of quantile functions below? $$ Q(p) = v_1 Q_1(p) + v_2 Q_2(p) = v_1 F_1^{-1}(p) + v_2 F_2^{-1}(p), \,\, v_1>0 , v_2 > 0,p\in[0,1] $$

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It's always hard to prove a negative. However, I have never seen such an interpretation. The naive idea that the weighted sum of quantiles would be the corresponding quantile of the mixture distribution is of course false.

I would be interested in being proven wrong and look forward to downvotes (and actual such interpretations).

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