### 统计代写|金融统计代写financial statistics代考| Random Vectors, Dependence, Correlation

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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

## 统计代写|金融统计代写financial statistics代考|Random Vectors, Dependence, Correlation

A random vector $\left(X_{1}, \ldots, X_{n}\right)$ from $\mathbb{R}^{n}$ can be useful in describing the mutual dependencies of several random variables $X_{1}, \ldots, X_{n}$, for example several underlying stocks. The joint distribution of the random variables $X_{1}, \ldots, X_{n}$ is as in the univariate case, uniquely determined by the probabilities
$$\mathrm{P}\left(a_{1} \leq X_{1} \leq b_{1}, \ldots, a_{n} \leq X_{n} \leq b_{n}\right), \quad-\infty<a_{i} \leq b_{i}<\infty, i=1, \ldots, n$$
If the random vector $\left(X_{1}, \ldots, X_{n}\right)$ has a density $p\left(x_{1}, \ldots, x_{n}\right)$, the probabilities can be computed by means of the following integrals:
$$\mathrm{P}\left(a_{1} \leq X_{1} \leq b_{1}, \ldots, a_{n} \leq X_{n} \leq b_{n}\right)=\int_{a_{n}}^{b_{n}} \ldots \int_{a_{1}}^{b_{1}} p\left(x_{1}, \ldots, x_{n}\right) d x_{1} \ldots d x_{n}$$
The univariate or marginal distribution of $X_{j}$ can be computed from the joint density by integrating out the variables not of interest.
$$\mathrm{P}\left(a_{j} \leq X_{j} \leq b_{j}\right)=\int_{-\infty}^{\infty} \cdots \int_{a_{j}}^{b_{j}} \cdots \int_{-\infty}^{\infty} p\left(x_{1}, \ldots, x_{n}\right) d x_{1} \ldots d x_{n}$$
The intuitive notion of independence of two random variables $X_{1}, X_{2}$ is formalized by requiring:
$$\mathrm{P}\left(a_{1} \leq X_{1} \leq b_{1}, a_{2} \leq X_{2} \leq b_{2}\right)=\mathrm{P}\left(a_{1} \leq X_{1} \leq b_{1}\right) \cdot \mathrm{P}\left(a_{2} \leq X_{2} \leq b_{2}\right),$$
i.e. the joint probability of two events depending on the random vector $\left(X_{1}, X_{2}\right)$ can be factorized. It is sufficient to consider the univariate distributions of $X_{1}$ and $X_{2}$ exclusively. If the random vector $\left(X_{1}, X_{2}\right)$ has a density $p\left(x_{1}, x_{2}\right)$, then $X_{1}$ and $X_{2}$ have densities $p_{1}(x)$ and $p_{2}(x)$ as well. In this case, independence of both random variables is equivalent to a joint density which can be factorized:
$$p\left(x_{1}, x_{2}\right)=p_{1}\left(x_{1}\right) p_{2}\left(x_{2}\right) .$$
Dependence of two random variables $X_{1}, X_{2}$ can be very complicated. If $X_{1}, X_{2}$ are jointly normally distributed, their dependency structure can be rather easily quantified by their covariance:
$$\operatorname{Cov}\left(X_{1}, X_{2}\right)=\mathrm{E}\left[\left(X_{1}-\mathrm{E}\left[X_{1}\right]\right)\left(X_{2}-\mathrm{E}\left[X_{2}\right]\right)\right],$$

## 统计代写|金融统计代写financial statistics代考|Conditional Probabilities and Expectations

The conditional probability that a random variable $Y$ takes values between $a$ and $b$ conditioned on the event that a random variable $X$ takes values between $x$ and $x+\Delta_{x}$, is defined as
$$\mathrm{P}\left(a \leq Y \leq b \mid x \leq X \leq x+\Delta_{x}\right)=\frac{\mathrm{P}\left(a \leq Y \leq b, x \leq X \leq x+\Delta_{x}\right)}{\mathrm{P}\left(x \leq X \leq x+\Delta_{x}\right)}$$
provided the denominator is different from zero. The conditional probability of events of the kind $a \leq Y \leq b$ reflects our opinion of which values are more plausible than others, given that another random variable $X$ has taken a certain value. If $Y$ is independent of $X$, the probabilities of $Y$ are not influenced by prior knowledge about $X$. It holds:
$$\mathrm{P}(a \leq Y \leq b \mid x \leq X \leq x+\Delta x)=\mathrm{P}(a \leq Y \leq b)$$

As $\Delta x$ goes to 0 in Eq. (3.4), the left side of Eq. (3.4) converges heuristically to $\mathrm{P}(a \leq Y \leq b \mid X=x)$. In the case of a continuous random variable $X$ having a density $p_{X}$, the left side of Eq. (3.4) is not defined since $\mathrm{P}(X=x)=0$ for all $x$. But, it is possible to give a sound mathematical definition of the conditional distribution of $Y$ given $X=x$. If the random variables $Y$ and $X$ have a joint distribution $p(x, y)$, then the conditional distribution has the density
$$p_{Y \mid X}(y \mid x)=\frac{p(x, y)}{p_{X}(x)} \quad \text { for } \quad p_{X}(x) \neq 0$$
and $p_{Y \mid X}(y \mid x)=0$ otherwise. Consequently, it holds:
$$\mathrm{P}(a \leq Y \leq b \mid X=x)=\int_{a}^{b} p_{Y \mid X}(y \mid x) d y$$
The expectation with respect to the conditional distribution can be computed by:
$$\mathrm{E}(Y \mid X=x)=\int_{-\infty}^{\infty} y_{p_{Y \mid X}}(y \mid x) d y \stackrel{\text { def }}{=} \eta(x)$$

## 统计代写|金融统计代写financial statistics代考|Recommended Literature

Exercise 3.1 Check that the random variable $X$ with $P(X=1)=1 / 2$, $P(X=-4)=1 / 3, P(X=5)=1 / 6$ has skewness 0 but is not distributed symmetrically.

Exercise $3.2$ Show that if $\operatorname{Cov}(X, Y)=0$ it does not imply that $X$ and $Y$ are independent.

Exercise $3.4$ We consider a bivariate exchange rates example, two European currencies, EUR and GBP, with respect to the USD. The sample period is $01 / 01 / 2002$ to $01 / 01 / 2009$ with altogether $n=1,828$ observations. Figure $3.1$ shows the time series of returns on both exchange rates.

Compute the correlation of the two exchange rate time series and comment on the sign of the correlation.

Exercise 3.5 Compute the conditional moments $\mathrm{E}\left(X_{2} \mid x_{1}\right)$ and $\mathrm{E}\left(X_{1} \mid x_{2}\right)$ for the $p d f$ of
$$f\left(x_{1}, x_{2}\right)=\left{\begin{array}{lc} \frac{1}{2} x_{1}+\frac{3}{2} x_{2} & 0 \leq x_{1}, x_{2} \leq 1 \ 0 & \text { otherwise } \end{array}\right.$$
Exercise $3.6$ Show that the function
$$f_{Y}\left(y_{1}, y_{2}\right)= \begin{cases}\frac{1}{2} y_{1}-\frac{1}{4} y_{2} & 0 \leq y_{1} \leq 2,\left|y_{2}\right| \leq 1-\left|1-y_{1}\right| \ 0 & \text { otherwise }\end{cases}$$
is a probability density function.
Exercise 3.7 Prove that $\mathrm{E} X_{2}=\mathrm{E}\left{\mathrm{E}\left(X_{2} \mid X_{1}\right)\right}$, where $\mathrm{E}\left(X_{2} \mid X_{1}\right)$ is the conditional expectation of $X_{2}$ given $X_{1}$.

Exercise 3.8 The conditional variance is defined as $\operatorname{Var}(Y \mid X)=\mathrm{E}[{Y-$ $\left.\mathrm{E}(Y \mid X)}^{2} \mid X\right]$. Show that $\operatorname{Var}(Y)=\mathrm{E}{\operatorname{Var}(Y \mid X)}+\operatorname{Var}{\mathrm{E}(Y \mid X)}$.

## 统计代写|金融统计代写financial statistics代考|Random Vectors, Dependence, Correlation

p(X1,X2)=p1(X1)p2(X2).

## 统计代写|金融统计代写financial statistics代考|Conditional Probabilities and Expectations

p是∣X(是∣X)=p(X,是)pX(X) 为了 pX(X)≠0

## 统计代写|金融统计代写financial statistics代考|Recommended Literature

$$f\left ( x_{1}, x_{2}\right)=\left{12X1+32X20≤X1,X2≤1 0 除此以外 \对。 和X和rC一世s和3.6小号H这在吨H一种吨吨H和F在nC吨一世这n f_{Y}\left(y_{1}, y_{2}\right)={12是1−14是20≤是1≤2,|是2|≤1−|1−是1| 0 除此以外$$

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