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

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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|NORMAL DISTRIBUTION

The normal distribution is probably the most widely used distribution in statistics, and is extremely popular in finance. The normal distribution occurs in a large number of settings, and is extremely easy to work with.

In popular literature, the normal distribution is often referred to as the bell curve because of the shape of its probability density function.

The probability density function of the normal distribution is symmetrical, with the mean and median coinciding with the highest point of the PDF. Because it is symmetrical, the skew of a normal distribution is always zero. The kurtosis of a normal distribution is always 3 . By definition, the excess kurtosis of a normal distribution is zero.

In some fields it is more common to refer to the normal distribution as the Gaussian distribution, after the famous German mathematician Johann Gauss, who is credited with some of the earliest work with the distribution. It is not the case that one name is more precise than the other as with mean and average. Both normal distribution and Gaussian distribution are acceptable terms.

For a random variable $X$, the probability density function for the normal distribution is:
$$f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-\beta}{\sigma}\right)^{2}}$$
The distribution is described by two parameters, $\mu$ and $\sigma ; \mu$ is the mean of the distribution and $\sigma$ is the standard deviation. We leave the proofs of these statements for the exercises at the end of the chapter.

Rather than writing out the entire density function, when a variable is normally distributed it is the convention to write:
$$X \sim N\left(\mu, \sigma^{2}\right)$$

This would be read ” $X$ is normally distributed with a mean of $\mu$ and variance of $\sigma^{2}$ “

One reason that normal distributions are easy to work with is that any linear combination of independent normal variables is also normal. If we have two normally distributed variables, $X$ and $Y$, and two constants, $a$ and $b$, then $Z$ is also normally distributed:
$$Z=a X+b Y \text { s.t. } Z \sim N\left(a \mu_{X}+b \mu_{Y}, a^{2} \sigma_{X}^{2}+b^{2} \sigma_{Y}^{2}\right)$$
This is very convenient. For example, if the log returns of individual stocks are normally distributed, then the average return of those stocks will also be normally distributed.

When a normal distribution has a mean of zero and a standard deviation of one, it is referred to as a standard normal distribution.
$$\phi=\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^{2}}$$
It is the convention to denote the standard normal PDF by $\phi$, and the cumulative standard normal distribution by $\Phi$.

Because a linear combination of normal distributions is also normal, standard normal distributions are the building blocks of many financial models. To get a normal variable with a standard deviation of $\sigma$ and a mean of $\mu$, we simply multiply the standard normal variable by $\sigma$ and add $\mu$.
$$X=\mu+\sigma \phi \Rightarrow X \sim N\left(\mu, \sigma^{2}\right)$$
To create two correlated normal variables, we can combine three independent standard normal variables, $X_{1}, X_{2}$, and $X_{3}$, as follows:
\begin{aligned} &X_{A}=\sqrt{\rho} X_{1}+\sqrt{1-\rho} X_{2} \ &X_{B}=\sqrt{\rho} X_{1}+\sqrt{1-\rho} X_{3} \end{aligned}

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

It’s natural to ask: if we assume that log returns are normally distributed, then how are standard returns distributed? To put it another way: rather than modeling log returns with a normal distribution, can we use another distribution and model standard returns directly?

The answer to these questions lies in the lognormal distribution, whose density function is given by:
$$f(x)=\frac{1}{x \sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{h x-\mu}{\sigma}\right)^{2}}$$
If a variable has a lognormal distribution, then the log of that variable has a normal distribution. So, if log returns are assumed to be normally distributed, then one plus the standard return will be lognormally distributed.
Unlike the normal distribution, which ranges from negative infinity to positive infinity, the lognormal distribution is undefined, or zero, for negative values. Given an asset with a standard return, $R$, if we model $(1+R)$ using the lognormal distribution, then $R$ will have a minimum value of $-100 \%$. As mentioned in Chapter 1 , this feature, which we associate with limited liability, is common to most financial assets. Using the lognormal distribution provides an easy way to ensure that we avoid returns less than $-100 \%$. The probability density function for a lognormal distribution is shown in Figure 4.6.

Equation $4.18$ looks almost exactly like the equation for the normal distribution, Equation $4.12$, with $x$ replaced by $\ln (x)$. Be careful, though, as there is also the $x$ in the denominator of the leading fraction. At first it might not be clear what the $x$ is doing there. By carefully rearranging Equation 4.18, we can get something that, while slightly longer, looks more like the normal distribution in form:

While not as pretty, this starts to hint at what we’ve actually done. Rather than being symmetrical around $\mu$, as in the normal distribution, the lognormal distribution is asymmetrical and peaks at $\exp \left(\mu-\sigma^{2}\right)$.

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|CENTRAL LIMIT THEOREMa

Assume we have an index made up of a large number of equities, or a bond portfolio that contains a large number of similar bonds. In these situations and many more, it is often convenient to assume that the constituent elements-the equities or bonds-are made up of statistically identical random variables, and that these variables are uncorrelated with each other. As mentioned previously, in statistics we term these variables independent and identically distributed (i.i.d.). If the constituent elements are i.i.d., it turns out we can say a lot about the distribution of the population, even if the distribution of the individual elements is unknown.

We already know that if we add two i.i.d. normal distributions together we get a normal distribution, but what happens if we add two i.i.d. uniform variables together? Looking at the graph of the uniform distribution (Figure 4.1), you might think that we would get another uniform distribution, but this isn’t the case. In fact, the probability density function resembles a triangle.

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## MATLAB代写

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