On Wed, 18 Apr 2001, Giuseppe Andrea Paleologo wrote:
> I am dealing with a simple conjecture. Given two generic positive
> random variables, is it always true that the sum of the quantiles (for
> a given value p) is greater or equal than the quantile of the sum?
>
< snip, technical translation of the question into algebra >
>
> Any insight or counterexample is greatly appreciated. I am sure this
> is proved in some textbook, but independently from that, I think this
> should be doable via elementary methods...
If this were a theorem, perhaps it should be. But it does not seem
inherently reasonable to me. (Herman Rubin has provided a mathematical
response denying the conjecture; but I'd like to look at it from a
different perspective. I'd be interested in opinions whether this line
of reasoning is valid.)
If I understand you correctly, you conjecture that for two random
variables (X and Y, say) and their sum (Z, say, = X + Y), the sum of the
third quartile of X and the third quartile of Y would be greater than or
equal to the third quartile of Z. But this would seem to imply, by
symmetry, that the sum of the _first_ quartile of X and the first
quartile of Y should be LESS than or equal to the first quartile of Z.
There being nothing especially magical about quartiles (whether first,
second, or third), these two statements together would imply that the
sum of a quantile of X and the corresponding quantile of Y must be BOTH
less than or equal to, AND greater than or equal to, the corresponding
quantile of Z: that is, the sum of the quantiles must always EQUAL the
corresponding quantile of the sum. But for this proposition, I believe
there exist lots of counterexamples.
-- DFB.
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Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
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