LECTURE 2: BASIC PROBABILITYTHEORYKey words: stochastic process, random experiment, sample space, event, Borel …eld, Sigmaalgebra, probability function, probability space, sets, union, intersection, complement.Required Textbook: The material in this lecture is related to Sections 2.1, 2.2, 2.3, 2.4, and2.5.Remark: Please read Chapter 1 of the textbook by yourself. This chapter gives some basicideas in statistics which are complement to our introduction. I will not cover it in class.1.1 Random ExperimentsDe…nition [Random Experiment]: A random experiment is a mechanism which has at leasttwo possible outcomes, and which outcome to occur is unknown in advance.Remark: The word “experiment”is used to mean a process that can generate some outcomes.It should be understood in a broader sense.Elements of a Random Experiment:(i) All possible outcomes.(ii) the probability with which each outcome will occur.Example 1 [Stock Price Changes] Suppose a stock can give two equally likely annual returns:30% and -10%. Compared to a bond with a 5% annual yied, will you like to invest your moneyin this stock or in bond?(a) There are at least two possible outcomes (two in this simple example); these outcomesare mutually exclusive in the sense that if one occurs, then the other will not occur.(b) Each year there is always one outcome coming out, but which one is uncertain. There aretwo possible outcomes (30% or -10%). Each outcome may occur with some chance (equal chancehere), but we do not which one will occur. We call the stock return, denoted by Y , a ”randomvariable”. It can take two values: 30% and -10%.Remark: Saying how many outcomes may occur is not su¢ cient; we must also say the chancewith which each outcome will occur.(c) The chance with which each outcome may occur will be called ”probability”. A higherprobability of an outcome implies more chance for the outcome to occur. We denote Pr(Y =30%)the probability that Y =30%, i.e. the chance with which Stock return is 30%. Similarly, wedenote Pr(Y =-10%) the probability that Y =-10%. ThenP(Y =30%)=0.5,P(Y =-10%)=0.5.1A complete description of Stock A is:Prob of Y 30% -10%0.50 0.50This is called the “probability distribution” of Y . In other words, the probability distributiongives a complete list of all possible outcomes with their chances of occurring.Question: How will an investor make a decision? Invest on stocks or bonds?Among other things, an investor will take into account two important factors: (i) expectedreturn, and (ii) risk. His/her decision will be a trade-o¤ between expected return and risk.Question: How to measure expected return and risk?Given the probability distribution of stock return Y , we are able to compute the so-calledexpected return and expected volatility: = E(Y ) =Xpiyi= 0:5  30% + 0:5  (10%) = 10%;2= var(Y ) =Xpi(yi )2= 0:5  (30%  10%)2+ 0:5  (10%  10%)2= (20%)2:Remarks:  and 2are called the mean and variance of random variable Y: They are the mostimportant objects in econometrics.Question: How will an investor make a decision based on  and 2? For this, we need to makesome assumptions on the economic behavior of the investor.Assume that the investor is risk adverse in the sense that he/she likes higher return but lowerrisk. That is, his/her utility function U(; 2) is a function of  and 2such that@U=@ > 0 : The more expected return, the better.@U=@2< 0 : The smaller risk, the better.An example of U(; 2) isU(; 2) = a  b2;where a > 0 and b > 0: The investor will maximize the utility function U(; 2) subject to thebudge constraint.In the current example, we assume the investor has totally I dollars to be split between sto ck(z) and saving (I  z). The total return, denoted by Y; has tow possible outcomes with equalprobability:2y = 1:30z + 1:05(I  z); ory = 0:90z + 1:05(I  z):It is easy to show that the mean  = 1:10z + 1:05(I  z) = 1:05I + 0:05z and the variance2= 0:04z2: The investor will solvemaxzU(; 2) = a  b2= a(1:05I + 0:05z)  b  0:04z2:The optimal investment will be z = 5a=8b: (Double check!)Remarks:(i) If b = 0 (i.e., the investor doesn’t care risk), then z = 1. All on the stock.(ii) If b = 1, then z = 0: All on saving.(ii) If 5a=8b < I; then z is an interior solution, which implies that the investor splits themoney between saving and stock.Question: What is meant if z > I? The investor will borrow money to invest in stock.Conclusion: Therefore, mean  and variance 2are two important factors for you to makea decision under uncertainty. In probability and statistics, both mean and variance are calledthe …rst moment and the second moment of random variable Y . They are the central focus ofstatistics and this course.(i) How to compute  and 2if possible outcomes may occur with di¤erent probabilities?(ii) How to update knowledge of  and 2if we have additional information X? For this,we need the so-called conditional mean E(Y jX) and conditional variance V ar (Y jX): These arecalled the regression function and volatility.For example, if X =interest rate, then E(Y jX) is the our knowledge of expected return of Ywhen we have information on X: Similarly for V ar(Y jX): This is useful to predict volatility ofY using the information of X: If Y is the volatility of another asset, V ar(Y jX) will allow us toinvestigate volatility spillover between assets or markets.These concepts are the most important concepts in this course. Of course, there also existsituations where  and  are not su¢ cient.A fundamental Axiom of Econometrics: (i) An economic system can be viewed as a randomexperiment governed by a probability law.(ii) Any economic phenomena (often in form of data) can be viewed as an outcome of thisrandom experiment. The random experiment is usually called a data generating process.Objective of econometrics: Inference of the probability law of the random experiment based onthe observed data.Remarks: There are alternative theories: chaos theory: an economic system is a deterministicprocess. For example, consider a chaotic logistic map:Yt= 4Yt1(1  Yt1);3where Y02 (0; 1): This is a pure deterministic process because if we know y0; then we can predictthe whole sequence surely. However, the data generated by this process is very similar to many…nancial time series data. Therefore, a natural question arise: is …nancial time series a