【正文】 ion i = 1 to infinity of. [P(x=xi) times (xi)]. I have it down that there might be an infinite number of possible values for the random variable x. In the case of the coin toss, there are only two, but I39。re accountable and we can list all possible values when they39。s called the expected value. People also call that the mean or the average. But, note that this is based on theory. These are probabilities. In order to pute using this formula you have to know the true probabilities. There39。s the same idea except thatI39。micro。s an integral. We have the integral from minus infinity to plus infinity of F(x)*x*dx, and that39。s the same thing because an integral is analogous to a summation. Those are the two population definitions. F(x) is the continuous probability distribution for x. That39。t have P (x = xi) because it39。 is zero because it could be 176。s an infinite number of possibilities. We have instead what39。re not going to need to know a lot about this for this course, but this isI wanted to get the basic ideas down. These are called population measures because they refer to the whole population of possible outes and they measure the probabilities. It39。s the summation i = 1 to n of xi/nthat39。m having a little trouble putting this into the Rituparna story, but you see the idea. You know the average, I assume. That39。s often reference to another kind of average, which I want to refer you to and which, in the Jeremy Siegel book, a lot is made of this. The other kind of average is called the geometric average. We39。ll only show the sample version of it G(x) = the product i = 1 to n of (xi )^(1/n). Does everyoneCan you see that? Instead of summing them and dividing by M, I multiply them all together and take the nth root of them. This is called the geometric average and it39。d have a problem, right? If you had one negative number in it, then the product would be a negative number and, if you took a root of that, then you might get an imaginary number. We don39。s an appendix to one of the chapters in Jeremy Siegel39。ve been investing money over a number of different years. Let39。s not a very good thing to do. What he says you should do instead is to take the geometric average of gross returns. The return on an investment is how much you made from the investment as a percent of the money invested. The gross return is the return plus one. The worst you can ever do investing is lose all of your investmentlose 100%. If we add one to the return, then you39。s never negative and we can then use geometric returns. Jeremy Siegel says that in finance we should be using geometric and not arithmetic averages. Why is that? Well I39。ve produced 20% a year for nine out of the last ten years. You think that39。s good. I would add up 20% a year for nine years and than put in a zero–no, 120 because it39。t look bad, right? But think about it, if you were investing your money with someone like that, what did you end up with? You ended up with nothing. If they have one year when they lose everything, it doesn39。s a less optimistic version. So, we should use that, but people in finance resist using that because it39。re advertising your return you want to make it look as big as possible. We also need some measure ofWe39。s often represented by σamp。, that39。sup2。sup2。micro。s the expectation of x or also E(x), so it39。s also another variance measure, which we use in the sampleor also Var is used sometimesand this is ∑amp。. There39。s just the summation i = 1 to n of (x x bar)amp。/n. That is the sample variance. Some people will divide by n–1. I suppose I would accept either answer. I39。 but I39。s a measure of how much x deviates from the mean。s squared. It weights big deviations a lot because the square of a big number is really big. So, that39。re going to be talking about these in finance in regards to returns becausegenerally the idea here is that we want high returns. We want a high expected value of returns, but we don39。s risk。s uncertainty. That39。s very basic here is covariance. Covariance is a measure of how much two variables move together. Covariance iswe39。ll just talk about it in a sample term. It39。re talking about an experiment when you generateEach experiment generates both an x and a y observation and we know when x is high, y also tends to be high, or whether it39。s a scaled covariance. We tend to use the Greek letter rho. If you were to use Excel, it would be correl or sometimes I say corr. That39。s the correlation coefficient. That has kind of almost entered the English language in the sense that you39。t know how much you39。s a high correlation. Does anyone know what it is? But you could estimate the corrit39。s way below one, but it has some correlation, so maybe it39。s very unlikely that it39。t be negative. It couldn39。ll give you a financial example. The concept of regression goes back to the mathematician Gauss, who talked about fitting a line through a scatter of points. Let39。s say Microsoft. I39。t put down a name of a pany because I can39。s not say Microsoft, let39。s no such pany, so I can be pletely hypothetical. Let39。re often negative. Suppose that in a given yearand say this is minus five and this is plus five, this is minus five and this is plus fiveSuppose that in the first year in our sample, the pany Shiller, Inc. and the market both did 5%. That puts a point right there at five and five. In another year, however, the stock market lost 5% and Shiller, Inc. lost 7%. We would have a point, say, down here at five and seven. This could be 1979, this could be 1980, and we keep adding points so we have a whole scatter of points. It39。s fit a line through the pointthe scatter of pointsand that39。s called the regression line and the intercept is called alphathere39。ve written it, the beta of Shiller, Inc. is the slope of this line. The alpha is just the intercept of this curve. We can also do this with excess returns. I will get to this later, where I have the return minus the interest rate on this axis and the market r