### 数学代写|计量经济学原理代写Principles of Econometrics代考|Expected Values of Several Random Variables

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|计量经济学原理代写Principles of Econometrics代考|Covariance Between Two Random Variables

The covariance between $X$ and $Y$ is a measure of linear association between them. Think about two continuous variables, such as height and weight of children. We expect that there is an association between height and weight, with taller than average children tending to weigh more than the average. The product of $X$ minus its mean times $Y$ minus its mean is
$$\left(X-\mu_{X}\right)\left(Y-\mu_{Y}\right)$$
In Figure P.4, we plot values ( $x$ and $y$ ) of $X$ and $Y$ that have been constructed so that $E(X)=E(Y)=0$.

The $x$ and $y$ values of $X$ and $Y$ fall predominately in quadrants I and III, so that the arithmetic average of the values $\left(x-\mu_{X}\right)\left(y-\mu_{Y}\right)$ is positive. We define the covariance between two random variables as the expected (population average) value of the product in (P.17).
$$\operatorname{cov}(X, Y)=\sigma_{X Y}=E\left[\left(X-\mu_{X}\right)\left(Y-\mu_{Y}\right)\right]=E(X Y)-\mu_{X} \mu_{Y}$$
We use the letters “cov” to represent covariance, and $\operatorname{cov}(X, Y)$ is read as “the covariance between $X$ and $Y$,” where $X$ and $Y$ are random variables. The covariance $\sigma_{X Y}$ of the random variables underlying Figure P.4 is positive, which tells us that when the values $x$ are greater

than $\mu_{X}$, then the values $y$ also tend to be greater than $\mu_{Y}$; and when the values $x$ are below $\mu_{X}$, then the values $y$ also tend to be less than $\mu_{Y}$. If the random variables values tend primarily to fall in quadrants II and IV, then $\left(x-\mu_{X}\right)\left(y-\mu_{Y}\right)$ will tend to be negative and $\sigma_{X Y}$ will be negative. If the random variables values are spread evenly across the four quadrants, and show neither positive nor negative association, then the covariance is zero. The sign of $\sigma_{X Y}$ tells us whether the two random variables $X$ and $Y$ are positively associated or negatively associated.

Interpreting the actual value of $\sigma_{X Y}$ is difficult because $X$ and $Y$ may have different units of measurement. Scaling the covariance by the standard deviations of the variables eliminates the units of measurement, and defines the correlation between $X$ and $Y$
$$\rho=\frac{\operatorname{cov}(X, Y)}{\sqrt{\operatorname{var}(X)} \sqrt{\operatorname{var}(Y)}}=\frac{\sigma_{X Y}}{\sigma_{X} \sigma_{Y}}$$
As with the covariance, the correlation $\rho$ between two random variables measures the degree of linear association between them. However, unlike the covariance, the correlation must lie between $-1$ and 1. Thus, the correlation between $X$ and $Y$ is 1 or $-1$ if $X$ is a perfect positive or negative linear function of $Y$. If there is no linear association between $X$ and $Y$, then $\operatorname{cov}(X, Y)=0$ and $\rho=0$. For other values of correlation the magnitude of the absolute value $|\rho|$ indicates the “strength” of the linear association between the values of the random variables. In Figure P.4, the correlation between $X$ and $Y$ is $\rho=0.5$.

## 数学代写|计量经济学原理代写Principles of Econometrics代考|Conditional Variance

The unconditional variance of a discrete random variable $X$ is
$$\operatorname{var}(X)=\sigma_{X}^{2}=E\left[\left(X-\mu_{X}\right)^{2}\right]=\sum_{x}\left(x-\mu_{X}\right)^{2} f(x)$$
It measures how much variation there is in $X$ around the unconditional mean of $X_{\text {, }} \mu_{X}$. For example, the unconditional variance $\operatorname{var}(W A G E)$ measures the variation in $W A G E$ around the unconditional mean $E(W A G E)$. In (P.13) we show that equivalently
$$\operatorname{var}(X)=\sigma_{X}^{2}=E\left(X^{2}\right)-\mu_{X}^{2}=\sum_{x} x^{2} f(x)-\mu_{X}^{2}$$
In Section P.6.1 we discussed how to answer the question “What is the mean wage of a person who has 16 years of education?” Now we ask “How much variation is there in wages for a person who has 16 years of education?* The answer to this question is given by the conditional variance, $\operatorname{var}(W A G E \mid E D U C=16)$. The conditional variance measures the variation in WAGE around the conditional mean $E(W A G E \mid E D U C=16)$ for individuals with 16 years of education. The conditional variance of WAGE for individuals with 16 years of education is the average squared difference in the population between WAGE and the conditional mean of WAGE,
$$\underbrace{\operatorname{var}(W A G E \mid E D U C=16)}{\text {conditional variance }}=E\left{[\underbrace{E D A G E-\underbrace{E(W A G E \mid E D U C=16)}}{\text {conditional mean }}]^{2} \mid E D U=16\right}$$
To obtain the conditional variance we modify the definitions of variance in equations (P.26) and (P.27); replace the unconditional mean $E(X)=\mu_{X}$ with the conditional mean $E(X \mid Y=y)$, and the unconditional $p d f f(x)$ with the conditional $p d f f(x \mid y)$. Then
$$\operatorname{var}(X \mid Y=y)=E\left{[X-E(X \mid Y=y)]^{2} \mid Y=y\right}=\sum_{x}(x-E(X \mid Y=y))^{2} f(x \mid y)$$
or
$$\operatorname{var}(X \mid Y=y)=E\left(X^{2} \mid Y=y\right)-[E(X \mid Y=y)]^{2}=\sum_{x} x^{2} f(x \mid y)-[E(X \mid Y=y)]^{2}$$

## 数学代写|计量经济学原理代写Principles of Econometrics代考|Proof of the Law of Iterated Expectations

Proof of the Law of Iterated Expectations To prove the Law of Iterated Expectations we make use of relationships between joint, marginal, and conditional $p d f$ s that we introduced in Section P.3. In Section P.3.1 we discussed marginal distributions. Given a joint $p d f$ $f(x, y)$ we can obtain the marginal pdf of $y$ alone $f_{Y}(y)$ by summing, for each $y$, the joint $p d f$ $f(x, y)$ across all values of the variable we wish to eliminate, in this case $x$. That is, for $Y$ and $X$,
\begin{aligned} &f(y)=f_{Y}(y)=\sum_{x} f(x, y) \ &f(x)=f_{X}(x)=\sum_{y} f(x, y) \end{aligned}
Because $f()$ is used to represent $p d f$ s in general, sometimes we will put a subscript, $X$ or $Y$, to be very clear about which variable is random.
Using equation (P.4) we can define the conditional pdf of $y$ given $X=x$ as
$$f(y \mid x)=\frac{f(x, y)}{f_{X}(x)}$$
Rearrange this expression to obtain
$$f(x, y)=f(y \mid x) f_{X}(x)$$
A joint $p d f$ is the product of the conditional $p d f$ and the $p d f$ of the conditioning variable.

To show that the Law of Iterated Expectations is true ${ }^{8}$ we begin with the definition of the expected value of $Y$, and operate with the summation.
\begin{aligned} E(Y) &=\sum_{y} y f(y)=\sum_{y} y\left[\sum_{x} f(x, y)\right] & & {[\text { substitute for } f(y)] } \ &=\sum_{y} y\left[\sum_{x} f(y \mid x) f_{X}(x)\right] & & {[\text { substitute for } f(x, y)] } \ &=\sum_{x}\left[\sum_{y} y f(y \mid x)\right] f_{X}(x) & & \text { [change order of summation] } \ &=\sum_{x} E(Y \mid x) f_{X}(x) & & {[\text { recognize the conditional expectation] }} \ &=E_{X}[E(Y \mid X)] & & \end{aligned}
While this result may seem an esoteric oddity it is very important and widely used in modern econometrics.

## 数学代写|计量经济学原理代写Principles of Econometrics代考|Covariance Between Two Random Variables

(X−μX)(是−μ是)

ρ=这⁡(X,是)曾是⁡(X)曾是⁡(是)=σX是σXσ是

## 数学代写|计量经济学原理代写Principles of Econometrics代考|Conditional Variance

\underbrace{\operatorname{var}(W A G E \mid E D U C=16)}{\text {条件方差}}=E\left{[\underbrace{E D A G E-\underbrace{E(W A G E \mid E D U C=16)} }{\text {条件均值}}]^{2} \mid E D U=16\right}\underbrace{\operatorname{var}(W A G E \mid E D U C=16)}{\text {条件方差}}=E\left{[\underbrace{E D A G E-\underbrace{E(W A G E \mid E D U C=16)} }{\text {条件均值}}]^{2} \mid E D U=16\right}

\operatorname{var}(X \mid Y=y)=E\left{[XE(X \mid Y=y)]^{2} \mid Y=y\right}=\sum_{x}(xE( X \mid Y=y))^{2} f(x \mid y)\operatorname{var}(X \mid Y=y)=E\left{[XE(X \mid Y=y)]^{2} \mid Y=y\right}=\sum_{x}(xE( X \mid Y=y))^{2} f(x \mid y)

## 数学代写|计量经济学原理代写Principles of Econometrics代考|Proof of the Law of Iterated Expectations

F(是)=F是(是)=∑XF(X,是) F(X)=FX(X)=∑是F(X,是)

F(是∣X)=F(X,是)FX(X)

F(X,是)=F(是∣X)FX(X)

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