3. Random Numbers and Monte Carlo Methods

• (Pseudo) Random numbers (see Numerical Recipes Section 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)
        a = numpy.random.random(10)
        b = numpy.random.uniform(-1, 1, 10)
        print(a)
        print(b)

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

• Generating probability distributions

  • Inverting the distribution (analytically or numerically) (see Numerical Recipes Section 7.2)
  • The rejection method in arbitrary dimensions (see Numerical Recipes Section 7.3).
    
  • A simple application.

  • Exercise 3.1: reproducing a given distribution   [Solution]

• Introduction to Monte-Carlo Techniques
  • (From Numerical Recipes)  Monte-Carlo Integration of functions

  • Calculating pi
  • Calculating an irregular area

  • Exercise 3.2   [Solution]