Drawing random numbers
The seed
Uniformly distributed random numbers
Visualizing the distribution
Vectorized drawing of random numbers
Computing the mean and standard deviation
The Gaussian or normal distribution
Drawing integers
Random integer functions
Example: Throwing a die
Drawing a random element from a list
Example: Drawing cards from a deck
Example: Class implementation of a deck
Computing probabilities
Principles of Monte Carlo simulation
Example: Throwing dice
Example: Drawing balls from a hat
Random mutations of genes
Example: Policies for limiting population growth
Simple games
Guessing a number
Rolling two dice
Monte Carlo integration
Derivation of Monte Carlo integration
Implementation of standard Monte Carlo integration
Area computing by throwing random points
Random walk in one space dimension
Basic implementation
Visualization
Random walk as a difference equation
Computing statistics of the particle positions
Vectorized implementation
Random walk in two space dimensions
Basic implementation
Vectorized implementation
Summary
Chapter topics
Example: Random growth
Exercises
Exercise 1: Flip a coin times
Exercise 2: Compute a probability
Exercise 3: Choose random colors
Exercise 4: Draw balls from a hat
Exercise 5: Computing probabilities of rolling dice
Exercise 6: Estimate the probability in a dice game
Exercise 7: Compute the probability of hands of cards
Exercise 8: Decide if a dice game is fair
Exercise 9: Adjust a game to make it fair
Exercise 10: Make a test function for Monte Carlo simulation
Exercise 11: Generalize a game
Exercise 12: Compare two playing strategies
Exercise 13: Investigate strategies in a game
Exercise 14: Investigate the winning chances of some games
Exercise 15: Compute probabilities of throwing two dice
Exercise 16: Vectorize flipping a coin
Exercise 17: Vectorize a probablility computation
Exercise 18: Throw dice and compute a small probability
Exercise 19: Is democracy reliable as a decision maker?
Exercise 20: Difference equation for random numbers
Exercise 21: Make a class for drawing balls from a hat
Exercise 22: Independent versus dependent random numbers
Exercise 23: Compute the probability of flipping a coin
Exercise 24: Simulate binomial experiments
Exercise 25: Simulate a poker game
Exercise 26: Estimate growth in a simulation model
Exercise 27: Investigate guessing strategies
Exercise 28: Vectorize a dice game
Exercise 29: Compute \( \pi \) by a Monte Carlo method
Exercise 30: Compute \( \pi \) by a Monte Carlo method
Exercise 31: Compute \( \pi \) by a random sum
Exercise 32: 1D random walk with drift
Exercise 33: 1D random walk until a point is hit
Exercise 34: Simulate making a fortune from gaming
Exercise 35: Simulate pollen movements as a 2D random walk
Exercise 36: Make classes for 2D random walk
Exercise 37: 2D random walk with walls; scalar version
Exercise 38: 2D random walk with walls; vectorized version
Exercise 39: Simulate mixing of gas molecules
Exercise 40: Simulate slow mixing of gas molecules
Exercise 41: Guess beer brands
Exercise 42: Simulate stock prices
Exercise 43: Compute with option prices in finance
Exercise 44: Differentiate noise measurements
Exercise 45: Differentiate noisy signals
Exercise 46: Model noise in a time signal
Exercise 47: Speed up Markov chain mutation
References
Random numbers have many applications in science and computer programming, especially when there are significant uncertainties in a phenomenon of interest. The purpose of this document is to look at some practical problems involving random numbers and learn how to program with such numbers. We shall make several games and also look into how random numbers can be used in physics. You need to be familiar with basic programming concepts such as loops, lists, arrays, vectorization, curve plotting, and command-line arguments in order to study the present document. A few examples and exercises will require familiarity with the class concept in Python.
The key idea in computer simulations with random numbers is first to formulate an algorithmic description of the phenomenon we want to study. This description frequently maps directly onto a quite simple and short Python program, where we use random numbers to mimic the uncertain features of the phenomenon. The program needs to perform a large number of repeated calculations, and the final answers are "only" approximate, but the accuracy can usually be made good enough for practical purposes. Most programs related to the present document produce their results within a few seconds. In cases where the execution times become large, we can vectorize the code. Vectorized computations with random numbers is definitely the most demanding topic in this document, but is not mandatory for seeing the power of mathematical modeling via random numbers.
All files associated with the examples in this document are found in the folder src/random.