### 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|PROBABILISTIC INEQUALITIES

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• Foundations of Data Science 数据科学基础

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Chebyshev’s Inequality

Some of the topics discussed in the book concern a setting in which we are not aware of the particular distribution of a random variable or the particular joint probability distribution of a pair of random variables. In such cases, the analysis may require us to resort to general arguments based on certain general inequalities from the theory of probability. In this section, we give an account of such inequalities and provide illustration where possible.

Chebyshev’s inequality provides a way to estimate the approximate probability of deviation of a random variable from its mean. Its most simple form concerns positive random variables.

Suppose that $X$ is a positive random variable, $X>0$. The following inequality is known as Chebyshev’s inequality,
$$P(X \geq \epsilon) \leq \frac{E X}{\epsilon},$$
where $\epsilon>0$. In this form, equation $(1.5)$ can be used to estimate the probability of observing a large observation by means of the mathematical expectation and the level $\epsilon$. Chebyshev’s inequality is rough as demonstrated geometrically in the following way. The mathematical expectation of a positive continuous random variable admits the representation,
$$E X=\int_{0}^{\infty} P(X \geq x) d x,$$

which means that it equals the area closed between the distribution function and the upper limit of the distribution function. This area is illustrated in Figure $1.9$ as the shaded area above the distribution function. On the other hand, the quantity $\epsilon P(X \geq \epsilon)=\epsilon\left(1-F_{X}(x)\right)$ is equal to the area of the rectangle in the upper-left corner of Figure $1.9$. In effect, the inequality
$$\epsilon P(X \geq \epsilon) \leq E X$$
admits the following geometric interpretation-the area of the rectangle is smaller than the shaded area in Figure 1.9.

For an arbitrary random variable, Chebychev’s inequality takes the form
$$P\left(|X-E X| \geq \epsilon \sigma_{X}\right) \leq \frac{1}{\epsilon^{2}}$$
where $\sigma_{X}$ is the standard deviation of $X$ and $\epsilon>0$. We use Chebyshev’s inequality in Chapter 6 in the discussion of dispersion measures.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Frechet-Hoeffding Inequality

Consider an $n$-dimensional random vector $Y$ with a distribution function $F_{Y}\left(y_{1}, \ldots, y_{n}\right)$. Denote by
$$W\left(y_{1}, \ldots, y_{n}\right)=\max \left(F_{Y_{1}}\left(y_{1}\right)+\cdots+F_{Y_{n}}\left(y_{n}\right)+1-n, 0\right)$$

and by
$$M\left(y_{1}, \ldots, y_{n}\right)=\min \left(F_{Y_{1}}\left(y_{1}\right), \ldots, F_{Y_{n}}\left(y_{n}\right)\right)$$
in which $F_{Y_{i}}\left(y_{i}\right)$ stands for the distribution function of the $i$-th marginal. The following inequality is known as Fréchet-Hoeffding inequality,
$$W\left(y_{1}, \ldots, y_{n}\right) \leq F_{Y}\left(y_{1}, \ldots, y_{n}\right) \leq M\left(y_{1}, \ldots, y_{n}\right) .$$
The quantities $W\left(y_{1}, \ldots, y_{n}\right)$ and $M\left(y_{1}, \ldots, y_{n}\right)$ are also called the Fréchet lower bound and the Fréchet upper bound. We apply FréchetHoeffding inequality in the two-dimensional case in Chapter 3 when discussing minimal probability metrics.

Since copulas are essentially probability distributions defined on the unit hypercube, Fréchet-Hoeffding inequality holds for them as well. In this case, it has a simpler form because the marginal distributions are uniform. The lower and the upper Fréchet bounds equal
and
$$\begin{gathered} W\left(u_{1}, \ldots, u_{n}\right)=\max \left(u_{1}+\cdots+u_{n}+1-n, 0\right) \ M\left(u_{1}, \ldots, u_{n}\right)=\min \left(u_{1}, \ldots, u_{n}\right) \end{gathered}$$
respectively. Fréchet-Hoeffding inequality is given by
$$W\left(u_{1}, \ldots, u_{n}\right) \leq C\left(u_{1}, \ldots, u_{n}\right) \leq M\left(u_{1}, \ldots, u_{n}\right) .$$
In the two-dimensional case, the inequality reduces to
$$\max \left(u_{1}+u_{2}-1,0\right) \leq C\left(u_{1}, u_{2}\right) \leq \min \left(u_{1}, u_{2}\right) .$$
In the two-dimensional case only, the lower Fréchet bound, sometimes referred to as the minimal copula, represents perfect negative dependence between the two random variables. In a similar way, the upper Fréchet bound, sometimes referred to as the maximal copula, represents perfect positive dependence between the two random variables.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|SUMMARY

We considered a number of concepts from probability theory that will be used in later chapters in this book. We discussed the notions of a random variable and a random vector. We considered one-dimensional and multidimensional probability density and distributions functions, which completely characterize a given random variable or random vector. We discussed statistical moments and quantiles, which represent certain characteristics of a random variable, and the sample moments which provide a way of estimating the corresponding characteristics from historical data. In the multidimensional case, we considered the notion of dependence between the components of a random vector. We discussed the covariance matrix versus the more general concept of a copula function. Finally, we described two probabilistic inequalities, Chebychev’s inequality and Fréchet-Hoeffding inequality.

ε磷(X≥ε)≤和X

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