### 统计代写|金融统计代写Mathematics with Statistics for Finance代考|Distributions

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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Mathematics with Statistics for Finance代考|PARAMETRIC DISTRIBUTIONS

Distributions can be divided into two broad categories: parametric distributions and nonparametric distributions. A parametric distribution can be described by a mathematical function. In the following sections we will explore a number of parametric distributions including the uniform distribution and the normal distribution. A nonparametric distribution cannot be summarized by a mathematical formula. In its simplest form, a nonparametric distribution is just a collection of data. An example of a nonparametric distribution would be a collection of historical returns for a security.

Parametric distributions are often easier to work with, but they force us to make assumptions, which may not be supported by real-world data. Nonparametric distributions can fit the observed data perfectly. The drawback of nonparametric distributions is that they are potentially too specific, which can make it difficult to draw any general conclusions.

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|UNIFORM DISTRIBUTION

For a continuous random variable, $X$, recall that the probability of an outcome occurring between $b_{1}$ and $b_{2}$ can be found by integrating as follows:
$$P\left[b_{1} \leq X \leq b_{2}\right]=\int_{b_{1}}^{b_{2}} f(x) d x$$
where $f(x)$ is the probability density function (PDF) of $X$.

The uniform distribution is one of the most fundamental distributions in statistics. The probability density function is given by the following formula:
$$u\left(b_{1}, b_{2}\right)=\left{\begin{array}{ll} c & \forall b_{1} \leq x \leq b_{2} \ 0 & \forall b_{1}>x>b_{2} \end{array} \quad \text { s.t. } b_{2}>b_{1}\right.$$
In other words, the probability density is constant and equal to $c$ between $b_{1}$ and $b_{2}$, and zero everywhere else. Figure $4.1$ shows the plot of a uniform distribution’s probability density function.

Because the probability of any outcome occurring must be one, we can find the value of $c$ as follows:
\begin{aligned} &\int_{-\infty}^{+\infty} u\left(b_{1}, b_{2}\right) d x=1 \ &\int_{-\infty}^{+\infty} u\left(b_{1}, b_{2}\right) d x=\int_{-\infty}^{b_{1}} 0 d x+\int_{b_{1}}^{b_{2}} c d x+\int_{b_{2}}^{+\infty} 0 d x=\int_{b_{1}}^{b_{2}} c d x \ &\int_{b_{1}}^{b_{2}} c d x=[\mathrm{cx}]{b{1}}^{b_{2}}=c\left(b_{2}-b_{1}\right)=1 \ &c=\frac{1}{b_{2}-b_{1}} \end{aligned}

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|BERNOULLI DISTRIBUTION

Bernoulli’s principle explains how the flow of fluids or gases leads to changes in pressure. It can be used to explain a number of phenomena, including how the wings of airplanes provide lift. Without it, modern aviation would be impossible. Bernoulli’s principle is named after Daniel Bernoulli, an eighteenthcentury Dutch-Swiss mathematician and scientist. Daniel came from a family of accomplished mathematicians. Daniel and his cousin Nicolas Bernoulli first described and presented a proof for the St. Petersburg Paradox. But it is not Daniel or Nicolas, but rather their uncle, Jacob Bernoulli, for whom the Bernoulli distribution is named. In addition to the Bernoulli distribution, Jacob is credited with first describing the concept of continuously compounded returns, and, along the way, discovering Euler’s number, $e$, both of which we explored in Chapter $1 .$

The Bernoulli distribution is incredibly simple. A Bernoulli random variable is equal to either zero or one. If we define $p$ as the probability that $X$ equals one, we have:
$$P[X=1]=p \text { and } P[X=0]=1-p$$
We can easily calculate the mean and variance of a Bernoulli variable:
\begin{aligned} \mu &=p \cdot 1+(1-p) \cdot 0=p \ \sigma^{2} &=p \cdot(1-p)^{2}+(1-p) \cdot(0-p)^{2}=p(1-p) \end{aligned}
Binary outcomes are quite common in finance: a bond can default or not default; the return of a stock can be positive or negative; a central bank can decide to raise rates or not to raise rates.

In a computer simulation, one way to model a Bernoulli variable is to start with a standard uniform variable. Conveniently, both the standard uniform variable and our Bernoulli probability, $p$, range between zero and one. If the draw from the standard uniform variable is less than $p$, we set our Bernoulli variable equal to one; likewise, if the draw is greater than $p$, we set the Bernoulli variable to zero (see Figure 4.2).

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|UNIFORM DISTRIBUTION

$$u\left(b_{1}, b_{2}\right)=\left{C∀b1≤X≤b2 0∀b1>X>b2\quad \text { st } b_{2}>b_{1}\right.$$

\begin{aligned} &\int_{-\infty}^{+\infty} u\left(b_{1}, b_{2}\right) dx=1 \ &\int_{-\ infty}^{+\infty} u\left(b_{1}, b_{2}\right) dx=\int_{-\infty}^{b_{1}} 0 d x+\int_{b_{1} }^{b_{2}} cd x+\int_{b_{2}}^{+\infty} 0 dx=\int_{b_{1}}^{b_{2}} cdx \ &\int_{b_{ 1}}^{b_{2}} cdx=[\mathrm{cx}] {b {1}}^{b_{2}}=c\left(b_{2}-b_{1}\right)= 1 \ &c=\frac{1}{b_{2}-b_{1}} \end{对齐}

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|BERNOULLI DISTRIBUTION

μ=p⋅1+(1−p)⋅0=p σ2=p⋅(1−p)2+(1−p)⋅(0−p)2=p(1−p)

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## MATLAB代写

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