金融市場的風險管理(英文版)(doc 14頁)

金融市場的風險管理(英文版)內容提要：
For discrete random variables, we can define the expected value, or µx --that's the Greek letter mu--as the summation 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 I'm saying in general there could be an infinite number. But they're accountable and we can list all possible values when they're discrete and form a probability weighted average of the outcomes. That'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 compute using this formula you have to know the true probabilities. There's another formula that applies for a continuous random variables and it's the same idea except that--I'll also call it µx, except that it's an integral. We have the integral from minus infinity to plus infinity of F(x)*x*dx, and that's really--you see it'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. That's different when you have continuous values--you don't have P (x = xi) because it's always zero. The probability that the temperature is exactly 100° is zero because it could be 100.0001° or something else and there's an infinite number of possibilities. We have instead what's called a probability density when we have continuous random variables. You're not going to need to know a lot about this for this course, but this is--I wanted to get the basic ideas down. These are called population measures because they refer to the whole population of possible outcomes and they measure the probabilities. It's the truth, but there are also sample means. When you get--this is Rituparna, counting the leaves on a tree--you can estimate, from a sample, the population expected values. The population mean is often written "x-bar." If you have a sample with n observations, it's the summation i = 1 to n of xi/n--that's the average. You know that formula, right? You count n leaves--you count the number of leaves. You have n branches on the tree and you count the number of leaves and sum them up. One would be--I'm having a little trouble putting this into the Rituparna story, but you see the idea. You know the average, I assume. That's the most elementary concept and you could use it to estimate either a discreet or continuous expected value.

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