How to sample points randomly below a curve? [duplicate]

Question

How can I randomly select 1000 points uniformly from the shaded area below the plotted curve?

Plot[1/π Cos[θ]^2, {θ, 0, 2 π}, Filling -> Bottom]

Answer 1

As noted in my comment, one approach is as follows. First, generate thousands of pairs of random numbers in the range {0, 2 π}, {0, 1/π}. Then select the first 1000 that lie below the curve.

lst = Transpose@{RandomReal[{0, 2 π}, 4000], RandomReal[{0, 1/π}, 4000]};
listsel = Select[lst, #[[2]] < 1/π Cos[#[[1]]]^2 &, 1000];
Show[Plot[1/π Cos[θ]^2, {θ, 0, 2 π}, Filling -> Bottom], ListPlot[listsel]]

This simple process works well provided the portion of points selected is a reasonable fraction of the total number of points, as it is here.

Answer 2

There is no need in filtering out random points in a rectangle that don't fall in the prescribed region. The sampling within a region can be done directly with RandomPoint.

Specify the region:

reg = ImplicitRegion[0 <= x <= 2 Pi && 0 <= y <= 1/Pi Cos[x]^2, {x, y}]

and then one can sample a point with

RandomPoint[reg]

e.g., {0.39486, 0.0422331}

or several points n with RandomPoint[reg, n]. There's a warning about an unbounded region, so to keep it clean one can add bounds as a third argument to RegionPlot:

data = RandomPoint[reg, 1000, {{0, 2 Pi}, {0, 1/Pi}}];

Show[RegionPlot[reg], ListPlot[data, Frame -> True], AspectRatio -> 1/GoldenRatio]

---

EDIT as per Trilarion's comment:

How does RandomPoint work internally is beyond my knowledge, but the timing analysis shows that it does not sample a rectangle and throw away the points that don't fall in the region (and even if it does, it's a way faster implementation):

reg = ImplicitRegion[0 <= x <= 10 && y >= x && y <= x + 1, {x, y}]

bbgodfrey's Select method:

n = 110000;
(lst = Transpose@{RandomReal[{0, 10}, n], RandomReal[{0, 11}, n]};
  sel = Select[lst, #[[1]] <= #[[2]] <= #[[1]] + 1 &, 
    UpTo[10000]];) // AbsoluteTiming
Length@sel

{0.221189, Null}

9916

10000 points weren't even generated.

RandomPoint approach:

pts = RandomPoint[reg, 10000, {{0, 10}, {0, 11}}]; // AbsoluteTiming

{0.049927, Null}

More than 4x faster, and all 10000 desired points are obviously generated.

Answer 3

More of a first principles approach, use the function as a PDF to generate random x data, then for each x choose a uniformly distributed point on the vertical line {x, f[x]}:

f[x_] := 1/π Cos[x]^2
z = Integrate[f[x], {x, 0, 2 π}]; (*can use NIntegrate here if needed*)
Plot[f[x], {x, 0, 2 π}, 
 Epilog -> 
  Point[
   {#, First@RandomVariate[UniformDistribution[{0, f[#]}], 1]} & /@ 
    RandomVariate[ProbabilityDistribution[f[x]/z, {x, 0, 2 π}], 1000]
  ]
]

This is likely the fastest approach.

I think newer versions of ProbabilityDistribution may do that normalization (/z) automatically, btw.

Answer 4

In case where only the plot is given:

plot = Plot[1/π Cos[θ]^2, {θ, 0, 2 π}, Filling -> Bottom]

polygons = Cases[plot // Normal, _Polygon, ∞]
region = RegionUnion @@ polygons;
pts = RandomPoint[region, 100]; (*quite slow*)
Show[plot, Graphics@Point@pts]