Generating Random Numbers

• (Pseudo) Random numbers (see Numerical Recipes Sections 7.0 and 7.1)

  • Random numbers in Python:

        import random
        random.seed(123)
        for i in range(10):
            print(random.random())
            print(random.uniform(-1, 1))

  • Random numbers in numpy:

        import numpy
        numpy.random.seed(123)
        x = numpy.random.random(10)
        y = numpy.random.uniform(-1, 1, 10)
        print(x)
        print(y)

• Generating points from probability distributions

  • Uniformly distributed points (see Numerical Recipes Section 7.1).
    • N points uniformly distributed in [a, b]:

          x = numpy.random.uniform(a, b, N)

    • N points uniformly distributed in a square:

          x = numpy.random.uniform(-1, 1, N)
          y = numpy.random.uniform(-1, 1, N)

      A simple application: 2-D uniform distribution. Here's an animated version: ran2d_anim.py

    • N points uniformly distributed in a cube:

          x = numpy.random.uniform(-1, 1, N)
          y = numpy.random.uniform(-1, 1, N)
          z = numpy.random.uniform(-1, 1, N)

  • The transformation method in one dimension (see Numerical Recipes Section 7.3.2).

  • The rejection method in arbitrary dimensions (see Numerical Recipes Section 7.3.6).

    • Example 1: Reproducing a distribution

    • Example 2: Uniform distribution in a sphere

  • Uniform distribution on the surface of a sphere. The surface element on a sphere has

        dA = sinθ dθ dφ = -dμ dφ

    where θ and φ are angular coordinates in spherical polar coordinates and μ = cosθ. Hence, to sample points on the sphere uniformly, we choose μ uniformly between -1 and 1 and φ uniformly between 0 and 2π:

        costh = numpy.random.uniform(-1, 1, N)
        phi = 2*numpy.pi*numpy.random.uniform(0, 1, N)
    
        sinth = numpy.sqrt(numpy.maximum(0, 1-costh**2))
        x = R*sinth*numpy.cos(phi)
        y = R*sinth*numpy.sin(phi)
        z = R*costh

    Here is a script that does this and displays the results.