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

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

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 (support@wolfram.com). 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]

Comments

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]

plotting - Adding a thick curve to a regionplot

Suppose we have the following simple RegionPlot: f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}] Now I'm trying to change the curve defined by $y=g[x]$ into a thick black curve, while leaving all other boundaries in the plot unchanged. I've tried adding the region $y=g[x]$ and playing with the plotstyle, which didn't work, and I've tried BoundaryStyle, which changed all the boundaries in the plot. Now I'm kinda out of ideas... Any help would be appreciated! Answer With f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 You can use Epilog to add the thick line: RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}, PlotPoints -> 50, Epilog -> (Plot[g[x], {x, 0, 2}, PlotStyle -> {Black, Thick}][[1]]), PlotStyle -> {Directive[Yellow, Opacity[0.4]], Directive[Pink, Opacity[0.4]],

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