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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

The concept of an average is standard in our culture. Pick up the newspaper and you will often see figures on average income, batting averages, or average crime rates. The concept of an average is intuitive. If someone earns $\$ 11,000$one year and$\$9,000$ in a second, we say his average income in the two years is $\$ 10,000$. If three children in a family are age 15,10 , and 5 , then we say the average age is 10 . In Table$4.1$the average return was$9 \%$. Statisticians usually use the term expected value to refer to what is commonly called an average. In this book we use both terms. An expected value or average is easy to compute. If all outcomes are equally likely, then to determine the average, one adds up the outcomes and divides by the number of outcomes. Thus, for Table$4.1$, the average is$(12+9+6) / 3=9$. A second way to determine an average is to multiply each outcome by the probability that it will occur. When the outcomes are not equally likely, this facilitates the calculation. Applying this procedure to Table$4.1$yields$\frac{1}{3}(12)+\frac{1}{3}(9)+\frac{1}{3}(6)=9$. It is useful to express this intuitive calculation in terms of a formula. The symbol$\Sigma$should be read “sum.” Underneath the symbol we put the first value in the sum and what is varying. On the top of the symbol we put the final value in the sum. We use the symbol$R_{i j}$to denote the$j$th possible outcome for the return on security$i$. Thus $$\frac{\sum_{j=1}^{3} R_{i j}}{3}=\frac{R_{a 1}+R_{j 2}+R_{b 3}}{3}=\frac{12+9+6}{3}$$ Using the summation notation just introduced and a bar over a variable to indicate expected return, we have for the expected value of the M equally likely returns for asset$i$: $$\bar{R}{i}=\sum{j=1}^{M} \frac{R_{i j}}{M}$$ If the outcomes are not equally likely and if$P_{i j}$is the probability of the$j$th return on the$i$th asset, then expected return is 1 $$\bar{R}{i}=\sum{j=1}^{M} P_{i j} R_{i j}$$ ## 金融代写|利率理论代写portfolio theory代考|A MEASURE OF DISPERSION Not only is it necessary to have a measure of the average return but it is also useful to have some measure of how much the outcomes differ from the average. The need for this second characteristic can be illustrated by the old story of the mathematician who believed an average by itself was an adequate description of a process and drowned in a stream with an average depth of 2 inches. Intuitively, a sensible way to measure how much the outcomes differ from the average is simply to examine this difference directly; that is, examine$R_{i j}-\bar{R}_{i}$. Having determined this for each outcome, one could obtain an overall measure by taking the average of this difference. Although this is intuitively sensible, there is a problem. Some of the differences will be positive and some negative, and these will tend to cancel out. The result of the canceling could be such that the average difference for a highly variable return need be no larger than the average difference for an asset with a highly stable return. In fact, it can be shown that the average value of this difference must always be precisely zero. The reader is encouraged to verify this with the example in Table 4.2. Thus the sum of the differences from the mean tells us nothing about dispersion. Two solutions to this problem suggest themselves. First, we could take absolute values of the difference between an outcome and its mean by ignoring minus signs when determining the average difference. Second, because the square of any number is positive, we could square all differences before determining the average. For ease of computation, when portfolios are considered, the latter procedure is generally followed. In addition, as we will see when we discuss utility functions, the average squared deviations have some convenient properties.${ }^{2}$The average squared deviation is called the variance; the square root of the variance is called the standard deviation. In Table$4.3$we present the possible returns from several hypothetical assets as well as the variance of the return on each asset. The alternative returns on any asset are assumed equally likely. Examining asset 1, we find the deviations of its returns from its average return are$(15-9),(9-9)$, and$(3-9)$. The squared deviations are 36,0 , and 36 , and the average squared deviation or variance is$(36+0+36) / 3=24$. To be precise, the formula for the variance of the return on the$i$th asset (which we symbolize as$\sigma_{l}^{2}$) when each return is equally likely is $$\sigma_{i}^{2}=\sum_{j=1}^{M} \frac{\left(R_{i j}-\bar{R}_{i}\right)^{2}}{M}$$ # 利率理论代考 ## 金融代写|利率理论代写portfolio theory代考|DETERMINING THE AVERAGE OUTCOME 平均值的概念在我们的文化中是标准的。拿起报纸，您会经常看到有关平均收入、打击率或平均犯罪率的数据。平均值的概念很直观。如果有人赚$\$11,000$ 一年 和 $\$ 9,000$在一秒钟内，我们说他两年的平均收入是$\$10,000$. 如果一个家庭的三个孩子分别是 $15 、 10$ 和 5 岁，那么我们说平均年龄是 10 岁。在表中 $4.1$ 平均回 报是 $9 \%$. 统计学家通常使用术语期望值来指代通常所说的平均值。在本书中，我们使用这两个术语。
㥍望值或平均值很容易计算。如果所有结果的可能性相同，那么为了确定平均值，将结果相加并除以结果的数量。因此，对于表 $4.1$ ，平均值为 $(12+9+6) / 3=9$. 确定平均值的第二种方法是将每个结果乘以它发生的概率。当结果的可能性不同时，这有助于计算。将此过程应用于表 $4.1$ 产量 $\frac{1}{3}(12)+\frac{1}{3}(9)+\frac{1}{3}(6)=9$ 放入总和中。我们使用符号 $R_{i j}$ 来表示 $j$ 安全回报的可能结果i. 因此
$$\frac{\sum_{j=1}^{3} R_{i j}}{3}=\frac{R_{a 1}+R_{j 2}+R_{b 3}}{3}=\frac{12+9+6}{3}$$

$$\bar{R} i=\sum j=1^{M} \frac{R_{i j}}{M}$$

$$\bar{R} i=\sum j=1^{M} P_{i j} R_{i j}$$

## 金融代写|利率理论代写portfolio theory代考|A MEASURE OF DISPERSION

$$\sigma_{i}^{2}=\sum_{j=1}^{M} \frac{\left(R_{i j}-\bar{R}_{i}\right)^{2}}{M}$$

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