Suppose Y is a mixture distribution defined as follows:


If 2 are beta distributed random variables (making Y a mixture of three beta distributions); and 3 are the 95th percentiles of the densities of Y and X1 respectively (similarly for X2 and X3), is it necessarily true that:


And does this hold for any rv's X1, X2, X3 that are defined over a positive range? Intuitively it makes sense that it does, but there may be something I haven't considered.


  • 2
    $\begingroup$ Consideration of cases where there are just two variables and one of them is certainly less than the other should quickly and clearly indicate why such a formula cannot possibly be correct. $\endgroup$
    – whuber
    Aug 8, 2017 at 17:05
  • $\begingroup$ Yes, you are of course correct (sorry, it's 3AM here) $\endgroup$
    – Anna Efron
    Aug 8, 2017 at 17:21

1 Answer 1


In general, NO.

Consider the simpler case of two different (independent) beta variables -- say a beta(1,2) and a beta(2,1). In that particular case a 50-50 mixture is standard uniform, which has 0.95 quantile equal to 0.95.

The weighted average of the two quantiles of the components is smaller than that (it's about 0.8755):

> qbeta(.95,1,2)
[1] 0.7763932
> qbeta(.95,2,1)
[1] 0.9746794

(This was the first example I tried, but knowing that quantiles are not linear in general, I expected almost any random example should fail unless it was a special case.)

[whuber's example in comments is an especially simple and compelling one -- but I wanted to show that it was still clearly the case for beta random variables - which overlap completely, since they're all on the unit interval]


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