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

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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|Population and Sample Data

If you wanted to know the mean age of people working in your firm, you would simply ask every person in the firm his or her age, add the ages together, and divide by the number of people in the firm. Assuming there are $n$ employees and $a_{i}$ is the age of the $i$ th employee, then the mean, $\mu$, is simply:
$$\mu=\frac{1}{n} \sum_{i=1}^{n} a_{i}=\frac{1}{n}\left(a_{1}+a_{2}+\cdots+a_{n-1}+a_{n}\right)$$
It is important at this stage to differentiate between population statistics and sample statistics. In this example, $\mu$ is the population mean. Assuming nobody lied about his or her age, and forgetting about rounding errors and

other trivial details, we know the mean age of people in your firm exactly. We have a complete data set of everybody in your firm; we’ve surveyed the entire population.

This state of absolute certainty is, unfortunately, quite rare in finance. More often, we are faced with a situation such as this: estimate the mean return of stock $\mathrm{ABC}$, given the most recent year of daily returns. In a situation like this, we assume there is some underlying data generating process, whose statistical properties are constant over time. The underlying process still has a true mean, but we cannot observe it directly. We can only estimate that mean based on our limited data sample. In our example, assuming $n$ returns, we estimate the mean using the same formula as before:
$$\hat{\mu}=\frac{1}{n} \sum_{i=1}^{n} r_{i}=\frac{1}{n}\left(r_{1}+r_{2}+\cdots+r_{n-1}+r_{n}\right)$$
where $\hat{\mu}$ (pronounced “mu hat”) is our estimate of the true mean based on our sample of $n$ returns. We call this the sample mean.

The median and mode are also types of averages. They are used less frequently in finance, but both can be useful. The median represents the center of a group of data; within the group, half the data points will be less than the median, and half will be greater. The mode is the value that occurs most frequently.

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|Discrete Random Variables

For a discrete random variable, we can also calculate the mean, median, and mode. For a random variable, $X$, with possible values, $x_{i}$, and corresponding probabilities, $p_{i}$, we define the mean, $\mu$, as:
$$\mu=\sum_{i=1}^{n} p_{i} x_{i}$$

The equation for the mean of a discrete random variable is a special case of the weighted mean, where the outcomes are weighted by their probabilities, and the sum of the weights is equal to one.

The median of a discrete random variable is the value such that the probability that a value is less than or equal to the median is equal to $50 \%$. Working from the other end of the distribution, we can also define the median such that $50 \%$ of the values are greater than or equal to the median. For a random variable, $X$, if we denote the median as $m$, we have:
$$P[X \geq m]=P[X \leq m]=0.50$$
For a discrete random variable, the mode is the value associated with the highest probability. As with population and sample data sets, the mode of a discrete random variable need not be unique.

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|Continuous Random Variables

We can also define the mean, median, and mode for a continuous random variable. To find the mean of a continuous random variable, we simply integrate the product of the variable and its probability density function (PDF). In the limit, this is equivalent to our approach to calculating the mean of a discrete random variable. For a continuous random variable, $X$, with a PDF, $f(x)$, the mean, $\mu$, is then:
$$\mu=\int_{x_{\min }}^{x_{\max }} x f(x) d x$$
The median of a continuous random variable is defined exactly as it is for a discrete random variable, such that there is a $50 \%$ probability that values are less than or equal to, or greater than or equal to, the median. If we define the median as $m$, then:
$$\int_{x_{\min }}^{m} f(x) d x=\int_{m}^{x_{\max }} f(x) d x=0.50$$
Alternatively, we can define the median in terms of the cumulative distribution function. Given the cumulative distribution function, $F(x)$, and the median, $m$, we have:
$$F(m)=0.50$$
The mode of a continuous random variable corresponds to the maximum of the density function. As before, the mode need not be unique.

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|Population and Sample Data

μ=1n∑一世=1n一种一世=1n(一种1+一种2+⋯+一种n−1+一种n)

μ^=1n∑一世=1nr一世=1n(r1+r2+⋯+rn−1+rn)

μ=∑一世=1np一世X一世

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|Continuous Random Variables

μ=∫X分钟X最大限度XF(X)dX

∫X分钟米F(X)dX=∫米X最大限度F(X)dX=0.50

F(米)=0.50

## 广义线性模型代考

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## MATLAB代写

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