# 数学代写|概率论代写Probability theory代考|THE MULTINOMIAL DISTRIBUTION

#### Doug I. Jones

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## 数学代写|概率论代写Probability theory代考|THE MULTINOMIAL DISTRIBUTION

The binomial distribution can easily be generalized to the case of $n$ repeated independent trials where each trial can have one of several outcomes. Denote the possible outcomes of each trial by $E_1, \ldots, E_r$, and suppose that the probability of the realization of $E_i$ in each trial is $p_i(i=1, \ldots, r)$. For $r=2$ we have Bernoulli trials; in general, the numbers $p_i$ are subject only to the condition
$$p_1+\cdots+p_r=1, \quad p_i \geq 0 .$$
The result of $n$ trials is a succession like $E_3 E_1 E_2 \ldots$ The probability that in $n$ trials $E_1$ occurs $k_1$ times, $E_2$ occurs $k_2$ times, etc., is
$$\frac{n !}{k_{1} ! k_{2} ! \cdots k_{r} !} p_1^{k_1} p_2^{k_2} p_3^{k_3} \cdots p_r^{k_r}$$
here the $k_i$ are arbitrary non-negative integers subject to the obvious condition
$$k_1+k_q+\cdots+k_r=n .$$
If $r=2$, then (9.2) reduces to the binomial distribution with $p_1=p$, $p_2=q, k_1=k, k_2=n-k$. The proof in the general case proceeds along the same lines, starting with II, (4.7).

Formula (9.2) is called the multinomial distribution because the righthand member is the general term of the multinomial expansion of $\left(p_1+\cdots+p_r\right)^n$. Its main application is to sampling with replacement when the individuals are classified into more than two categories (e.g., according to professions).

## 数学代写|概率论代写Probability theory代考|THE NORMAL DISTRIBUTION

In order to avoid later interruptions we pause here to introduce two functions of great importance.
Definition. The function defined by
$$\mathrm{n}(x)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^2}$$
is called the normal density function; its integral
$$\mathfrak{N}(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^x e^{-\frac{1}{2} \nu^2} d y$$
is the normal distribution function.
The graph of $\mathfrak{n}(x)$ is the symmetric, bell-shaped curve shown in figurc 1 . Note that different units are used along the two axes: The maximum of $\mathfrak{n}(x)$ is $1 / \sqrt{2 \pi}=0.399$, approximately, so that in an ordinary Cartesian system the curve $y=\mathrm{n}(x)$ would be much flatter. [The notations $\mathrm{n}$ and $\mathfrak{N}$ are not standard. In the first two editions the more customary $\phi$ and $\Phi$ were used, but in volume 2 consistency required that we reserve these letters for other purposes.]

Lemma 1. The domain bounded by the graph of $\mathrm{n}(x)$ and the $x$-axis has unit area, that is,
$$\int_{-\infty}^{+\infty} n(x) d x=1$$
Proof. We have
\begin{aligned} \left{\int_{-\infty}^{+\infty} n(x) d x\right}^2 & =\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} n(x) n(y) d x d y= \ & =\frac{1}{2 \pi} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} e^{-\frac{1}{2}\left(x^2+y^2\right)} d x d y . \end{aligned}
This double integral can be expressed in polar coordinates thus:
$$\frac{1}{2 \pi} \int_0^{2 \pi} d \theta \int_0^{\infty} e^{-\frac{1}{2} r^2} r d r=\int_0^{\infty} e^{-\frac{1}{2} r^2} r d r=-\left.e^{-\frac{1}{2} r^2}\right|_0 ^{\infty}=1$$
which proves the assertion.
It follows from the definition and the lemma that $\mathfrak{N}(x)$ increases steadily from 0 to 1. Its graph (figure 2) is an S-shaped curve with
$$\mathfrak{N}(-x)=1-\mathfrak{N}(x)$$

# 概率论代考

## 数学代写|概率论代写Probability theory代考|THE MULTINOMIAL DISTRIBUTION

$$p_1+\cdots+p_r=1, \quad p_i \geq 0 .$$
$n$试验的结果是一个类似$E_3 E_1 E_2 \ldots$的连续，在$n$试验中$E_1$发生$k_1$次，$E_2$发生$k_2$次等的概率为
$$\frac{n !}{k_{1} ! k_{2} ! \cdots k_{r} !} p_1^{k_1} p_2^{k_2} p_3^{k_3} \cdots p_r^{k_r}$$

$$k_1+k_q+\cdots+k_r=n .$$

## 数学代写|概率论代写Probability theory代考|THE NORMAL DISTRIBUTION

$$\mathrm{n}(x)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} x^2}$$

$$\mathfrak{N}(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^x e^{-\frac{1}{2} \nu^2} d y$$

$\mathfrak{n}(x)$的图形是如图1所示的对称钟形曲线。请注意，沿着两个轴使用不同的单位:$\mathfrak{n}(x)$的最大值近似为$1 / \sqrt{2 \pi}=0.399$，因此在普通的笛卡尔系统中，曲线$y=\mathrm{n}(x)$会平坦得多。[注:$\mathrm{n}$和$\mathfrak{N}$不标准。]在前两个版本中使用了更习惯的$\phi$和$\Phi$，但在第二卷中，为了保持一致性，我们将这些字母保留到其他用途。

$$\int_{-\infty}^{+\infty} n(x) d x=1$$

\begin{aligned} \left{\int_{-\infty}^{+\infty} n(x) d x\right}^2 & =\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} n(x) n(y) d x d y= \ & =\frac{1}{2 \pi} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} e^{-\frac{1}{2}\left(x^2+y^2\right)} d x d y . \end{aligned}

$$\frac{1}{2 \pi} \int_0^{2 \pi} d \theta \int_0^{\infty} e^{-\frac{1}{2} r^2} r d r=\int_0^{\infty} e^{-\frac{1}{2} r^2} r d r=-\left.e^{-\frac{1}{2} r^2}\right|_0 ^{\infty}=1$$

$$\mathfrak{N}(-x)=1-\mathfrak{N}(x)$$

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MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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