Random variables with transformed discrete distributions cannot be applied to Probability[] function?

Question

I have two discrete uniform distributions over same support:

Dx = DiscreteUniformDistribution[{-8, 8}];  
Dy = DiscreteUniformDistribution[{-8, 8}];

I use random variables X $\sim$ Dx, Y $\sim$ Dy to model condition in some if-else clause in C code. For example,

if (3 * x + 1 > y) {
   ... 
}

So, I model this condition by random event 3*X + 1 > Y. Then probability I look for is:

Probability[
  x > y, 
  {Distributed[x, TransformedDistribution[3 * x + 1, Distributed[x, Dx]]],
   Distributed[y, Dy]}
]

Unfortunately, Mathematica produces nothing. As for me, this is very strange behavior because Mathematica also cannot compute mean for the product distribution of {X, Y}. Maybe, this problem is caused by different support of X and Y.

How can I compute this probability?

Answer 1

The following computes the probability you asked for, but I'm at a loss to understand why your notation won't work. It seems correct to me.

Probability[
 3 x + 1 > y, {Distributed[x, Dx], Distributed[y, Dy]}]

(*
==> 147/289
*)

The Probability / Distribution functions are quite young and hence not fully bulletproofed and I have encountered a number of bugs myself. I suppose/hope that in the next release they will be solved. I suggest you contact Wolfram support ([email protected]). But perhaps someone else here may find out what's wrong.

A few of mine:

No random variates from PDFs that stem from multivariate distributions:

dist = ProbabilityDistribution[
   PDF[BinormalDistribution[{0, 0}, {1, 1}, 0]][{x, 
     y}], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}];
RandomVariate[dist]

This works for 3, but not for 5 (and higher):

Expectation[x,x\[Distributed] OrderDistribution[{GeometricDistribution[0.1], 3}, 3]]
Expectation[x,x\[Distributed] OrderDistribution[{GeometricDistribution[0.1], 5}, 5]]

The following crashes the kernel after a minute or so (so, DO NOT EXECUTE):

PDF[TransformedDistribution[Sin[u], u \[Distributed] NormalDistribution[0, 1]], 0]