# 统计代写|统计推断代写Statistical inference代考|STATS2107

## 统计代写|统计推断代写Statistical inference代考|Expectation and joint moments

We often encounter situations where we are interested in a function of several random variables. Consider the following illustration: let $X_1, \ldots, X_5$ represent our models for the total rainfall in December at five locations around the UK. Functions that may be of interest include:

• the mean across locations, $\frac{1}{5} \sum_{i=1}^5 X_i$.
• the maximum across locations, $\max _i\left(X_i\right)$.
• the mean of the four rainiest locations, $\frac{1}{4}\left[\sum_{i=1}^5 X_i-\min _i\left(X_i\right)\right]$.
As a function of random variables, each of these is itself a random variable. In the situations that we will consider, if $X_1, \ldots, X_n$ are random variables and $g: \mathbb{R}^n \rightarrow \mathbb{R}$ is a function of $n$ variables, then $g\left(X_1, \ldots, X_n\right)$ is also a random variable. In many instances, the distribution of $g\left(X_1, \ldots, X_n\right)$ is of interest; this topic is tackled in section 4.6. We start with something more straightforward: calculation of the mean.

The expected value of a function of two random variables is given by an extension of Theorem 3.4.4.
Proposition 4.3.1 (Expectation of a function of two random variables)
If $g$ is a well-behaved, real-valued function of two variables, $g: \mathbb{R}^2 \longrightarrow \mathbb{R}$, and $X$ and $Y$ are random variables with joint mass/density function $f_{X, Y}$, then
$$\mathbb{E}[g(X, Y)]= \begin{cases}\sum_y \sum_x g(x, y) f_{X, Y}(x, y) & \text { (discrete case) } \ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(x, y) f_{X, Y}(x, y) d x d y & \text { (continuous case) }\end{cases}$$
An example using the simple polynomial density function of $4.2 .10$ follows.
Example 4.3.2 (Expectation of a product for simple polynomial density) Consider random variables $X$ and $Y$ with joint density function $f_{X, Y}(x, y)=x+y$ for $0<x<1$ and $0<y<1$. Suppose we are interested in calculating the expectation of the product of $X$ and $Y$, that is, $\mathbb{E}(X Y)$. Using Proposition 4.3.1 with $g(X, Y)=X Y$ we have
\begin{aligned} \mathbb{E}(X Y) &=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x y f_{X, Y}(x, y) d x d y \ &=\int_0^1 \int_0^1 x y(x+y) d x d y \ &-\int_0^1\left[\frac{x^3}{3} y+\frac{x^2}{2} y^2\right]_{x=0}^{x=1} d y-\int_0^1\left(\frac{1}{3} y+\frac{1}{2} y^2\right) d y \ &=\left[\frac{1}{6} y^2+\frac{1}{6} y^3\right]_0^1=\frac{1}{3} \end{aligned}

## 统计代写|统计推断代写Statistical inference代考|Covariance and correlation

In the univariate case, we discussed the use of single-number summaries for the features of a distribution. For example, we might use the mean as a measure of central tendency and the variance as a measure of spread. In the multivariate case we might, in addition, be interested in summarising the dependence between random variables. For a pair of random variables, a commonly used quantity for measuring the degree of (linear) association is correlation. The starting point for the definition of correlation is the notion of covariance.
Definition 4.3.5 (Covariance)
For random variables $X$ and $Y$, the covariance between $X$ and $Y$ is defined as
$$\operatorname{Cov}(X, Y)=\mathbb{E}[(X-\mathbb{E}(X))(Y-\mathbb{E}(Y))] .$$
An alternative form for the covariance is
$$\operatorname{Cov}(X, Y)=\mathbb{E}(X Y)-\mathbb{B}(X) \mathbb{E}(Y) .$$
Proving the equivalence of these two forms is part of Exercise 4.3. Covariance has a number of properties that are immediate consequences of its definition as an expectation.
Claim 4.3.6 (Properties of covariance)
For random variables $X, Y, U$, and $V$ the covariance has the following properties.
i. Symmetry: $\operatorname{Cov}(X, Y)=\operatorname{Cov}(Y, X)$.

# 统计推断代考

## 统计代写|统计推断代写统计推断代考|期望和关节力矩

• 跨地点的平均值，$\frac{1}{5} \sum_{i=1}^5 X_i$ .
• 跨地点的最大值，$\max _i\left(X_i\right)$ .
• 四个多雨地点的平均值，$\frac{1}{4}\left[\sum_{i=1}^5 X_i-\min _i\left(X_i\right)\right]$ .
作为随机变量的函数，每一个它们本身都是一个随机变量。在我们将要考虑的情况下，如果$X_1, \ldots, X_n$是随机变量，$g: \mathbb{R}^n \rightarrow \mathbb{R}$是$n$变量的函数，那么$g\left(X_1, \ldots, X_n\right)$也是一个随机变量。在许多情况下，$g\left(X_1, \ldots, X_n\right)$的分布是令人感兴趣的;这个主题将在第4.6节中讨论。我们从更直接的开始:计算平均值。

$$\mathbb{E}[g(X, Y)]= \begin{cases}\sum_y \sum_x g(x, y) f_{X, Y}(x, y) & \text { (discrete case) } \ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(x, y) f_{X, Y}(x, y) d x d y & \text { (continuous case) }\end{cases}$$

\begin{aligned} \mathbb{E}(X Y) &=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x y f_{X, Y}(x, y) d x d y \ &=\int_0^1 \int_0^1 x y(x+y) d x d y \ &-\int_0^1\left[\frac{x^3}{3} y+\frac{x^2}{2} y^2\right]_{x=0}^{x=1} d y-\int_0^1\left(\frac{1}{3} y+\frac{1}{2} y^2\right) d y \ &=\left[\frac{1}{6} y^2+\frac{1}{6} y^3\right]_0^1=\frac{1}{3} \end{aligned}

## 统计代写|统计推断代写统计推断代考|协方差和相关性

.

$$\operatorname{Cov}(X, Y)=\mathbb{E}[(X-\mathbb{E}(X))(Y-\mathbb{E}(Y))] .$$

$$\operatorname{Cov}(X, Y)=\mathbb{E}(X Y)-\mathbb{B}(X) \mathbb{E}(Y) .$$

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