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Finding expected value over uncountable number of pseudo-random points?

Motivation

We wish to create a function that appears to be a "pseudo-randomly" distributed but has infinite points that are non-uniform (i.e. does not have complete spatial randomness) in the sub-space of $\mathbb{R}^2$, where the expected value or integral of the function w.r.t uniform probability measure is non-obvious (i.e. not the center of the space the function covers nor the area of that space).

Question 1: What procedure can be used to find the expected value, using the uniform probability measure, of such a distribution of points?

Example

Suppose we have a code where for real numbers $x_1,x_2,y_1$ and $y_2$, we generate an uncountable number of "nearly pseudo-random" points which are non-uniform in the subspace $[x_1,x_2]\times[y_1,y_2]\subseteq\mathbb{R}^2$. We represent this using the function f[x,k], where as $k$ grows larger; the points in the graph appears "pseudo-random" despite being non-uniformly distributed. (To visualize function $f$, I was forced to set $k$ to a low value; however, the graph below the code should convince one that I'm correct.)

b = 3 (*base b expansion of the x-values in [x1,x2]*)
x1 = 0
x2 = 1
y1 = 0
y2 = 1
k = 20
g1[xr_, r_] := 
 g1[xr, r] = 
  Round[(10/b) Sin[r xr] + (10/b)]

f[x_, t_] := 
 f[x, t] = 
  N[y2 - ((y2 - y1)/(10)) RealAbs[ 
      Sum[g1[Sum[RealDigits[x, b, t, -r][[1]][[z]], {z, r + 1, t}], 
          r + 1]/b^r, {r, 0, 8}] - 
       10]] 

p = .00005 (*Incremement between the x-values in the points of the
graph below*)

ListPlot[Table[{x, f[x, k]}, {x, x1 + p, x2, p}]] 
(*Graphs countable points of the functions but is not a
completely accurate graph. Here t=k=20, where the larger b is, the smaller k must
to be to "appears pseudo-random" on the countably many points.*)

enter image description here

Question 2: How do we find the expected value, w.r.t the uniform probability measure, of $f(x,20)$? (Note I only graphed countably many points; we must also include the remaining points when taking the expected value.)

I doubt we could compute the expected value mathematically; however, if I'm wrong let me know.

If the expected value doesn't exist or is non-finite, I found a way of manipulating the expected value so there exists a new expected value (see the attatchment).

(Optional) What is the expected value of f[x,20] using the attatchment?

Attachments:
POSTED BY: Bharath Krishnan
3 Replies

Its not quite clear what the distribution should be, so lets display the main examples:

  1. Let

    Z = Function[{r, z, phi}, {r Sqrt[1 - z^2] Cos[phi], r Sqrt[1 - z^2] Sin[phi], r z}]

be the coordinate function spherical -> cartesian in R^3 with z=cos theta.

By dz = d cos theta = sin theta dtheta the uniform distribution over the z-axis generates a uniform distribution over meridians ans ba this fact over the sphere

Let X the random variable uniform over {0,1} X: = RandomReal[];

The := definition generates a new value fo each function call.

then

Graphics3D[Point /@ Array[Z[10 + X, 2 X - 1, 2 Pi X ] &, {300}]]

generates a uniform spherical non-uniform radial distribution of points over a spherical shell {10,11}.

Sperical Scatter

In this way its possible to generate distributions ad libitum. Setting one variable constant yields surface distributions. Using complicated distributions with correlations can be generated by use of more complicated nonlinear functions of the three variables. Hidden variable distributions can be generated by addition of a fourth independent random variable and adding the same random value of the current event to the other variables.

  1. To generate distributions uniform with respect to the Lebesgue measure in R^2,3 scatter a volume with points {X,X,X} and select all points subject to a function

( a < f[#1, #2, #3] < b &)

Graphics3D[ Point /@  Select[Array[({2 X - 1, 2 X - 1,   2 X - 1} &), {20000}],
,(0.2 < #[[1]]^2 + #[[2]]^2 - #[[3]]^2 <  0.25 &)]]

HyperbolaScatter

POSTED BY: Roland Franzius

How would we find the expected value of these distributions?

Also, I need to make sure we have an uncountable number of pseudo-random points. Your distributions appears to not do that.

If there are several distributions of my example could you give an example of such a distribution for the uncountable points.

POSTED BY: Bharath Krishnan

Cross posted at https://mathematica.stackexchange.com/questions/283525/finding-expected-value-over-uncountable-number-of-pseudo-random-points-non-unif

Note "w.r.t uniform probability measure" means "use the uniform density to sample the nonuniform one in order to approximate a statistic".

POSTED BY: Bharath Krishnan
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