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1、Class 1: Expectations, variances, and basics of estimationBasics of matrix (1) I. Organizational Matters(1) Course requirements:1) Exercises: There will be seven (7) exercises, the last of which is optional. Each exercise will be graded on a scale of 0-10. In addition to the graded exercise, an answ
2、er handout will be given to you in lab sections. 2) Examination: There will be one in-class, open-book examination. (2) Computer software: StataII. Teaching Strategies(1) Emphasis on conceptual understanding. Yes, we will deal with mathematical formulas, actually a lot of mathematical formulas. But,
3、 I do not want you to memorize them. What I hope you will do, is to understand the logic behind the mathematical formulas. (2) Emphasis on hands-on research experience.Yes, we will use computers for most of our work. But I do not want you to become a computer programmer. Many people think they know
4、statistics once they know how to run a statistical package. This is wrong. Doing statistics is more than running computer programs. What I will emphasize is to use computer programs to your advantage in research settings. Computer programs are like automobiles. The best automobile is useless unless
5、someone drives it. You will be the driver of statistical computer programs. (3) Emphasis on student-instructor communication.I happen to believe in students judgment about their own education. Even though I will be ultimately responsible if the class should not go well, I hope that you will feel par
6、t of the class and contribute to the quality of the course. If you have questions, do not hesitate to ask in class. If you have suggestions, please come forward with them. The class is as much yours as mine. Now let us get to the real business.III(1). Expectation and VarianceRandom Variable: A rando
7、m variable is a variable whose numerical value is determined by the outcome of a random trial.Two properties: random and variable.A random variable assigns numeric values to uncertain outcomes. In a common language, give a number. For example, income can be a random variable. There are many ways to
8、do it. You can use the actual dollar amounts. In this case, you have a continuous random variable. Or you can use levels of income, such as high, median, and low. In this case, you have an ordinal random variable 1=high, 2=median, 3=low. Or if you are interested in the issue of poverty, you can have
9、 a dichotomous variable: 1=in poverty, 0=not in poverty.In sum, the mapping of numeric values to outcomes of events in this way is the essence of a random variable.Probability Distribution: The probability distribution for a discrete random variable X associates with each of the distinct outcomes xi
10、 (i = 1, 2,., k) a probability P(X = xi).Cumulative Probability Distribution: The cumulative probability distribution for a discrete random variable X provides the cumulative probabilities P(X x) for all values x.Expected Value of Random Variable: The expected value of a discrete random variable X i
11、s denoted by EX and defined:EX = P(xi)where: P(xi) denotes P(X = xi). The notation E (read “expectation of”) is called the expectation operator.In common language, expectation is the mean. But the difference is that expectation is a concept for the entire population that you never observe. It is the
12、 result of the infinite number of repetitions. For example, if you toss a coin, the proportion of tails should be .5 in the limit. Or the expectation is .5. Most of the times you do not get the exact .5, but a number close to it. Conditional ExpectationIt is the mean of a variable conditional on the
13、 value of another random variable. Note the notation: E(Y|X).In 1996, per-capita average wages in three Chinese cities were (in RMB):Shanghai:3,778Wuhan:1,709Xian: 1,155Variance of Random Variable: The variance of a discrete random variable X is denoted by VX and defined:VX = (xi - EX)2 P(xi)where:
14、P(xi) denotes P(X = xi). The notation V (read “variance of”) is called the variance operator.Since the variance of a random variable X is a weighted average of the squared deviations, (X - EX)2 , it may be defined equivalently as an expected value: VX = E(X - EX)2. An algebraically identical express
15、ion is: VX = EX2 - (EX)2.Standard Deviation of Random Variable: The positive square root of the variance of X is called the standard deviation of X and is denoted by sX: s X =The notation s (read “standard deviation of”) is called the standard deviation operator.Standardized Random Variables: If X is a random variable with expected value EX and standard deviation sX, then:Y=is known as the standardized form of random variable X. Covariance: The covariance of two discrete random variables X and Y
