ISTA 311 Programming assignment 1 Due: Thursday, September 26 (11:59 PM) Problem Statement A probability distribution can be represented in Python as a dictionary, where the keys are the possible outcomes and the values are their probabilities. For example, the dictionary {“heads”: 0.5, “tails”: 0.5} can be used to represent the probability distribution of a fair coin flip. For our purposes, we will define a class called Distribution that acts as a wrapper for a dictionary, defining several additional methods. A template for the class is on D2L, with the init method already defined. You do not need to modify the init method, but you should read it and understand it. To complete the assignment, implement the following methods: Calculating probability (10 points) Our first task is to calculate the probability of a set of outcomes. Write a method called prob. Your method should take one parameter in addition to self: a Python set object. It should return the real number between 0 and 1 which is the probability of the event corresponding to the set. (You may assume that the set does not contain any elements not in the dictionary; that is, you don’t need to write code protecting against KeyErrors.) Normalizing a distribution (15 points) In applying Bayes’ theorem later in the course, we will encounter intermediate steps where a distribution does not follow Kolmogorov’s laws because the sum of the probabilities is not 1. It will be useful to correct this. Write a method called normalize which takes only the parameter self. This method should modify the dictionary in-place by scaling the probabilities so that they add up to 1. You may return None. Conditioning a distribution (15 points) We have seen that the process of calculating conditional probabilities P (A|B) is equivalent to eliminating those outcomes inconsistent with the given event B, and recalculating probabilities so that they sum to 1 while preserving the relative sizes of those probabilities. Write a method called condition which takes one parameter in addition to self: a Python set object which is a subset of the keys of the dictionary (i.e., an event). Your method should modify the dictionary in-place by removing key-value pairs corresponding to outcomes not in the parameter set, and recalculating the probabilities so that they sum to 1. The end result is the conditional distribution, conditioned on the event defined by the parameter set. You may return None. Simulating a random process (20 points) Our second task is to simulate the experience whose probability distribution is described by a dictionary. Write a method called sample. Your method should take no parameters besides self. It should return a randomly chosen outcome (dictionary key) according to the probability distribution described by the dictionary. You may use any function that generates uniformly distributed random numbers from [0, 1) (such as random.random(), or numpy.random.uniform()), but no other source of randomness. You may assume that the dictionary passed as input will follow Kolmogorov’s laws: that is, the keys represent mutually exclusive outcomes and the values sum to 1. Test cases A Python module containing several test cases will be available on D2L. Note: this test script requires SciPy and matplotlib to run.
