### 统计代写|金融统计代写financial statistics代考| Basic Concepts of Probability Theory

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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|Real Valued Random Variables

Thanks to Newton’s laws, dropping a stone from a height of $10 \mathrm{~m}$, the point of time of its impact on the ground is known before executing the experiment. Quantities in complex systems (such as stock prices at a certain date, daily maximum temperature at a certain place) are, however, not deterministically predictable, although it is known which values are more likely to occur than others. Contrary to the falling stone, data which cannot be described successfully by deterministic mechanism can be modelled by random variables.

Let $X$ be such a random variable that (as a model for stock prices) takes values in real time. The appraisal of which values of $X$ are more and which are less likely is expressed by the probability of events as ${a<X<b}$ or ${X \leq b}$. The set of all probabilities
$$\mathrm{P}(a \leq X \leq b), \quad-\infty<a \leq b<\infty$$
determines the distribution of $X$. In other words, the distribution is defined by the probabilities of all events which depend on $X$. In the following, we denote the probability distribution of $X$ by $\mathcal{L}(X)$.

The probability distribution is uniquely defined by the cumulative probability distribution
$$F(x)=\mathrm{P}(X \leq x), \quad-\infty<x<\infty$$
$F(x)$ monotonously increases and converges for $x \rightarrow-\infty$ to 0 , and for $x \rightarrow \infty$ to 1. If there is a function $p$, such that the probabilities can be computed by means of an integral
$$\mathrm{P}(a<X<b)=\int_{a}^{b} p(x) d x$$

$p$ is called a probability density, or briefly density of $X$. Then the cumulative distribution function is a primitive of $p$ :
$$F(x)=\int_{-\infty}^{x} p(y) d y$$
For small $h$ it holds:
$$\mathrm{P}(x-h<X<x+h) \approx 2 h \cdot p(x)$$

## 统计代写|金融统计代写financial statistics代考|Expectation and Variance

The mathematical expectation or the mean $\mathrm{E}[X]$ of a real random variable $X$ is a measure for the location of distribution of $X$. Adding to $X$ a real constant $c$, it holds for the expectation: $\mathrm{E}[X+c]=\mathrm{E}[X]+c$, i.e. the location of the distribution is translated. If $X$ has a density $p(x)$, its expectation is defined as:
$$\mathrm{E}(X)=\int_{-\infty}^{\infty} x p(x) d x .$$
If the integral does not exist, neither does the expectation. In practice, this is rather rarely the case.

Let $X_{1}, \ldots, X_{n}$ be a sample of identically independently distributed (i.i.d.) random variables (see Sect. 3.4) having the same distribution as $X$, then $\mathrm{E}[X]$ can be estimated by means of the sample mean:
$$\hat{\mu}=\frac{1}{n} \sum_{t=1}^{n} X_{t} .$$
A measure for the dispersion of a random variable $X$ around its mean is given by the variance $\operatorname{Var}(X)$ :
\begin{aligned} \operatorname{Var}(X)=& \mathrm{E}\left[(X-\mathrm{E} X)^{2}\right] \ \text { Variance }=& \text { mean squared deviation of a random variable } \ & \text { around its expectation. } \end{aligned}
If $X$ has a density $p(x)$, its variance can be computed as follows:
$$\operatorname{Var}(X)=\int_{-\infty}^{\infty}(x-\mathrm{E} X)^{2} p(x) d x$$
The integral can be infinite. There are empirical studies giving rise to doubt that some random variables appearing in financial and actuarial mathematics and which model losses in highly risky businesses dispose of a finite variance.

As a quadratic quantity the variance’s unit is different from that of $X$ itself. It is better to use the standard deviation of $X$ which is measured in the same unity as $X$ :
$$\sigma(X)=\sqrt{\operatorname{Var}(X)}$$
Given a sample of i.i.d. variables $X_{1}, \ldots, X_{n}$ which have the same distribution as $X$, the sample variance can be estimated by:
$$\hat{\sigma}^{2}=\frac{1}{n} \sum_{t=1}^{n}\left(X_{t}-\hat{\mu}\right)^{2} .$$

## 统计代写|金融统计代写financial statistics代考|Skewness and Kurtosis

Definition 3.1 (Skewness) The skewness of a random variable $X$ with mean $\mu$ and variance $\sigma^{2}$ is defined as
$$S(X)=\frac{\mathrm{E}\left[(X-\mu)^{3}\right]}{\sigma^{3}}$$
If the skewness is negative (positive) the distribution is skewed to the left (right). Normally distributed random variables have a skewness of zero since the distribution is symmetrical around the mean. Given a sample of i.i.d. variables $X_{1}, \ldots, X_{n}$, skewness can be estimated by (see Sect. 3.4)
$$\hat{S}(X)=\frac{\frac{1}{n} \sum_{i=1}^{n}\left(X_{i}-\hat{\mu}\right)^{3}}{\hat{\sigma}^{3}}$$
with $\hat{\mu}, \hat{\sigma}^{2}$ as defined in the previous section.
Definition $3.2$ (Kurtosis) The kurtosis of a random variable $X$ with mean $\mu$ and variance $\sigma^{2}$ is defined as
$$\operatorname{Kurt}(X)=\frac{E\left[(X-\mu)^{4}\right]}{\sigma^{4}}$$
Normally distributed random variables have a kurtosis of 3 . Financial data often exhibits higher kurtosis values, indicating that values close to the mean and extreme positive and negative outliers appear more frequently than for normally distributed random variables. For i.i.d. sample kurtosis can be estimated by
$$\widehat{\operatorname{Kurt}}(X)=\frac{\frac{1}{n} \sum_{i=1}^{n}\left(X_{i}-\hat{\mu}\right)^{4}}{\hat{\sigma}^{4}} .$$

## 统计代写|金融统计代写financial statistics代考|Real Valued Random Variables

F(X)=磷(X≤X),−∞<X<∞
F(X)单调增加和收敛X→−∞为 0 ，并且对于X→∞1.如果有函数p, 这样概率可以通过积分来计算

p被称为概率密度，或简称密度X. 那么累积分布函数是p :
F(X)=∫−∞Xp(是)d是

## 统计代写|金融统计代写financial statistics代考|Expectation and Variance

μ^=1n∑吨=1nX吨.

σ(X)=曾是⁡(X)

σ^2=1n∑吨=1n(X吨−μ^)2.

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## MATLAB代写

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