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# 数学代写|概率论代考Probability Theory代写|Math461 Generating Functions

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## 数学代写|概率论代考Probability Theory代写|Generating Functions

In this section we have a special look at random variables that take values in $\mathbb{N}$. We shall introduce a new concept, the generating function of such a random variable. There are at least two good reasons for doing so.

In the first place, generating functions are a very convenient tool for all sorts of computations, that would be difficult and tedious without them. These computations have to do with sums of random variables, expectations and variances. For an application of this, see Section 6.5.

In the second place, they are a very good warm up for the concept of characteristic functions which are similar in nature but not restricted to integer-valued random variables. In this section, $X, Y, \ldots$ are random variables taking values in $\mathbb{N}$.
Definition 2.6.1. The generating function of $X$ is defined as
\begin{aligned} G_X(s) & =E\left(s^X\right) \ & =\sum_{n=0}^{\infty} p_X(n) s^n, \end{aligned}
for all $s \in \mathbb{R}$ for which this sum converges.

## 数学代写|概率论代考Probability Theory代写|Random Walk and Counting

Consider a particle on the one-dimensional line, starting at position $a \in \mathbb{Z}$, say. This particle performs $n$ random steps as follows. At each time $t=1,2, \ldots, n$, we flip a coin. If heads comes up, the particle moves one unit to the right, if tails comes up, the particle moves one unit to the left. The sample space corresponding to this experiment is simply $\Omega={-1,+1}^n$, the set of sequences of length $n$ with the symbols $-1$ and $+1$. For instance, if $n=4$, and $\omega=(1,1,-1,1)$, then the first and second step are to the right, the third is to the left, and the fourth is to the right again. The reason that we define $\Omega$ as ${-1,1}^n$ rather than ${0,1}^n$, is that in the former case, we can conveniently represent the position of the particle after $k$ steps by the random variable $S_k: \Omega \rightarrow \mathbb{R}$ as
$$S_k(\omega)=a+\sum_{i=1}^k \omega_i,$$
for $k=1,2, \ldots, n$. The probability measure on $\Omega$ associated to the random walk gives equal probability $2^{-n}$ to all $2^n$ possible outcomes. Hence, if we want to compute the probability of an event $A$, we need to count the number of elements in $A$ and multiply by $2^{-n}$. We should denote this probability measure by $P_n$, but we will drop the subscript $n$ when no confusion is possible. We will mostly be interested in the joint distribution of the $S_k$ ‘s, that is, in the random vector
$$S(\omega)=\left(S_1(\omega), \ldots, S_n(\omega)\right)$$

# 概率论代写

## 数学代写|概率论代考Probability Theory代写|Generating Functions

$$G_X(s)=E\left(s^X\right) \quad=\sum_{n=0}^{\infty} p_X(n) s^n,$$

## 数学代写|概率论代考Probability Theory代写|Random Walk and Counting

$$S_k(\omega)=a+\sum_{i=1}^k \omega_i,$$

$$S(\omega)=\left(S_1(\omega), \ldots, S_n(\omega)\right)$$

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## MATLAB代写

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。