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

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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|DISCRETE RANDOM VARIABLES

The concept of probability is central to risk management. Many concepts associated with probability are deceptively simple. The basics are easy, but there are many potential pitfalls.

In this chapter, we will be working with both discrete and continuous random variables. Discrete random variables can take on only a countable number of values-for example, a coin, which can only be heads or tails, or a bond, which can only have one of several letter ratings (AAA, AA, A, BBB, etc.). Assume we have a discrete random variable $X$, which can take various values, $x_{i}$. Further assume that the probability of any given $x_{i}$ occurring is $p_{i}$. We write:
$$P\left[X=x_{i}\right]=p_{i} \text { s.t. } x_{i} \in\left{x_{1}, x_{2}, \ldots, x_{n}\right}$$
where $P[\cdot]$ is our probability operator.”
An important property of a random variable is that the sum of all the probabilities must equal one. In other words, the probability of any event

occurring must equal one. Something has to happen. Using our current notation, we have:
$$\sum_{i=i}^{n} p_{i}=1$$

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|CONTINUOUS RANDOM VARIABLES

In contrast to a discrete random variable, a continuous random variable can take on any value within a given range. A good example of a continuous random variable is the return of a stock index. If the level of the index can be any real number between zero and infinity, then the return of the index can be any real number greater than $-1$.

Even if the range that the continuous variable occupies is finite, the number of values that it can take is infinite. For this reason, for a continuous variable, the probability of any specific value occurring is zero.

Even though we cannot talk about the probability of a specific value occurring, we can talk about the probability of a variable being within a certain range. Take, for example, the return on a stock market index over the next year. We can talk about the probability of the index return being between $6 \%$ and $7 \%$, but talking about the probability of the return being exactly $6.001 \%$ or exactly $6.002 \%$ is meaningless. Even between $6.001 \%$ and $6.002 \%$ there are literally an infinite number of possible values. The probability of any one of those infinite values occurring is zero.
For a continuous random variable $X$, then, we can write:
$$P\left[r_{1}<X<r_{2}\right]=p$$
which states that the probability of our random variable, $X$, being between $r_{1}$ and $r_{2}$ is equal to $p$.

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|Probability Density Functions

For a continuous random variable, the probability of a specific event occurring is not well defined, but some events are still more likely to occur than others. Using annual stock market returns as an example, if we look at 50 years of data, we might notice that there are more data points between $0 \%$ and $10 \%$ than there are between $10 \%$ and $20 \%$. That is, the density of points between $0 \%$ and $10 \%$ is higher than the density of points between $10 \%$ and $20 \%$.

For a continuous random variable we can define a probability density function (PDF), which tells us the likelihood of outcomes occurring between any two points. Given our random variable, $X$, with a probability $p$ of being between $r_{1}$ and $r_{2}$, we can define our density function, $f(x)$, such that:
$$\int_{r_{1}}^{r_{2}} f(x) d x=p$$
The probability density function is often referred to as the probability distribution function. Both terms are correct, and, conveniently, both can be abbreviated PDF.

As with discrete random variables, the probability of any value occurring must be one:
$$\int_{r_{\min }}^{r_{\max }} f(x) d x=1$$
where $r_{\min }$ and $r_{\max }$ define the lower and upper bounds of $f(x)$.

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|DISCRETE RANDOM VARIABLES

P\left[X=x_{i}\right]=p_{i} \text { st } x_{i} \in\left{x_{1}, x_{2}, \ldots, x_{n}\对}P\left[X=x_{i}\right]=p_{i} \text { st } x_{i} \in\left{x_{1}, x_{2}, \ldots, x_{n}\对}

∑一世=一世np一世=1

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|Probability Density Functions

∫r1r2F(X)dX=p

∫r分钟r最大限度F(X)dX=1

## 广义线性模型代考

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## MATLAB代写

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