Notes for Math 450 Lecture Notes 1

We often have to deal with situations involving hidden or unpredictable factors. Sometimes the unpredictability is an essential element of the phenomenon under investigation. For example, physicists regard radioactive decay as a purely random phenomenon caused by quantum fluctuations of the state of radioactive atoms. More often, the unpredictability is due to our lack of information of all the factors affecting the system, on account of its complexity or instability, as in weather forecasting. Or because information is deliberately being kept hidden from us, for example, when playing a game of poker. Even relatively simple, deterministic systems can be very unstable and behave in a practically unpredictable way. (See the discussion of the forced pendulum in a later lecture.) In such cases it may be fruitful to formulate a probabilistic model that can serve as the basis for forecasting or decision-making regarding the system. For example, although it is not practically possible to predict whether a person will enter or leave a public building at any given time, for most purposes we can treat the arrival times as random, with a probability distribution depending on a few parameters that can be estimated by statistical methods. Similarly, we cannot predict what the exchange rate between the dollar and the euro will be next week, but we may be able to probabilistically model its fluctuations well enough to estimate the likelihood that this rate will be within a given range of values. Sometimes we may want to use a randomized procedure to find the solution to a problem or to make a decision. There are many problems in mathematics in which a random search in a large set of candidates for a solution can be much more efficient than a systematic search. (This relates to the topic of Random Algorithms and Monte Carlo Methods.) Even in ordinary life situations we may want to contrive random procedures for deciding on a course of action, either because randomness may be equated with fairness (e.g., when picking which soccer team will give the initial kick) or because the situation at hand is so complicated that any attempt to deliberate in more analytical ways could lead to not making any decision at all (e.g., whether to marry Camilla or Sophia.) Throughout this course we will consider probabilistic models of a variety of phenomena in areas ranging from finance to molecular biology. We will learn