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How can I create a list of random pairs $(x,y)$ where $0<y<x<1$ and $x,y \in \mathbb{R}$ ? I can't seem to place that inequality restriction on the random numbers that are generated.

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RandomReal[1, {100, 2}] /. {x_, y_} /; y > x :> {y, x}

The graphics below confirm that you retain a uniform distribution

Mathematica graphics

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Generate random pairs, and reverse-sort each one:

Sort[#, Greater] & /@ RandomReal[1, {10, 2}]

or, for fun, generate two numbers between 0 and 0.5, and add the second to the first:

RandomReal[0.5, {10, 2}].{{1, 0}, {1, 1}}
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    $\begingroup$ I don't believe the second method is valid: the smaller value can never exceed 0.5, incorrectly ruling out e.g. {0.8, 0.7}. $\endgroup$ – Mr.Wizard Jul 25 '14 at 20:41
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    $\begingroup$ Note: While the second method has problems the first is correct, and it is four times faster than RunnyKine's (admittedly pretty) method. $\endgroup$ – Mr.Wizard Jul 26 '14 at 9:52
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Another way is

{#, RandomReal[#]} & /@   RandomVariate[TriangularDistribution[{0, 1}, 1], 1000]

To see we do have a uniform distribution:

PDF[
    TransformedDistribution[{x, y}, {
                x  TriangularDistribution[{0, 1}, 1],
                y  UniformDistribution[{0, b}]
                }] /. b -> x,
    {x, y}]

uniform distribution on 0<y<x<1

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    $\begingroup$ It looks like that is NOT an uniform distribution. $\endgroup$ – Silvia Jul 25 '14 at 20:12
  • $\begingroup$ @Silvia yes it is not uniformly distributed $\endgroup$ – Algohi Jul 26 '14 at 3:51
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    $\begingroup$ Maybe something like {#, RandomReal[#]} & /@ RandomVariate[TriangularDistribution[{0, 1}, 1], 100000]? $\endgroup$ – Silvia Jul 26 '14 at 8:08
  • $\begingroup$ @Silvia Very good. Either edit this Answer to include that or post your own. $\endgroup$ – Mr.Wizard Jul 26 '14 at 9:51
  • $\begingroup$ @Mr.Wizard done :) $\endgroup$ – Silvia Jul 26 '14 at 11:25

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