在概率论概念中，随机过程是随机变量的集合。 若一随机系统的样本点是随机函数，则称此函数为样本函数，这一随机系统全部样本函数的集合是一个随机过程。 实际应用中，样本函数的一般定义在时间域或者空间域。 随机过程的实例如股票和汇率的波动、语音信号、视频信号、体温的变化，随机运动如布朗运动、随机徘徊等等。
A random process can be viewed as a function $X(t, \omega)$ of two variables, time $t \in \mathcal{T}$ and the outcome of the underlying random experiment $\omega \in \Omega$
– For fixed $t, X(t, \omega)$ is a random variable over $\Omega$
– For fixed $\omega, X(t, \omega)$ is a deterministic function of $t$, called a sample function
fdas
Discrete Time Random Process
- A random process is said to be discrete time if $\mathcal{T}$ is a countably infinite set, e.g.,
० $\mathcal{N}={0,1,2, \ldots}$
$\circ \mathbf{Z}={\ldots,-2,-1,0,+1,+2, \ldots}$ - In this case the process is denoted by $X_{n}$, for $n \in \mathcal{N}$, a countably infinite set, and is simply an infinite sequence of random variables
- A sample function for a discrete time process is called a sample sequence or sample path
- A discrete-time process can comprise discrete, continuous, or mixed r.v.s
Specifying a Random Process
In the above examples we specified the random process by describing the set of sample functions (sequences, paths) and explicitly providing a probability measure over the set of events (subsets of sample functions)
This way of specifying a random process has very limited applicability, and is suited only for very simple processes
A random process is typically specified (directly or indirectly) by specifying all its $n$-th order cdfs (pdfs, pmfs), i.e., the joint cdf (pdf, pmf) of the samples
$$
X\left(t_{1}\right), X\left(t_{2}\right), \ldots, X\left(t_{n}\right)
$$
for every order $n$ and for every set of $n$ points $t_{1}, t_{2}, \ldots, t_{n} \in \mathcal{T}$
The following examples of important random processes will be specified (directly or indirectly) in this manner
Important Classes of Random Processes
- IID process: $\left{X_{n}: n \in \mathcal{N}\right}$ is an IID process if the r.v.s $X_{n}$ are i.i.d. Examples:
- Bernoulli process: $X_{1}, X_{2}, \ldots, X_{n}, \ldots$ i.i.d. $\sim \operatorname{Bern}(p)$
- Discrete-time white Gaussian noise (WGN): $X_{1}, \ldots, X_{n}, \ldots$ i.i.d. $\sim \mathcal{N}(0, N)$
- Here we specified the $n$-th order pmfs (pdfs) of the processes by specifying the first-order pmf (pdf) and stating that the r.v.s are independent
- It would be quite difficult to provide the specifications for an IID process by specifying the probability measure over the subsets of the sample space
The Random Walk Process
- Let $Z_{1}, Z_{2}, \ldots, Z_{n}, \ldots$ be i.i.d., where
$$
Z_{n}= \begin{cases}+1 & \text { with probability } \frac{1}{2} \ -1 & \text { with probability } \frac{1}{2}\end{cases}
$$ - The random walk process is defined by
$$
\begin{aligned}
&X_{0}=0 \
&X_{n}=\sum_{i=1}^{n} Z_{i}, \quad n \geq 1
\end{aligned}
$$ - Again this process is specified by (indirectly) specifying all $n$-th order pmfs
- Sample path: The sample path for a random walk is a sequence of integers, e.g.,
$$
0,+1,0,-1,-2,-3,-4, \ldots
$$
or
$$
0,+1,+2,+3,+4,+3,+4,+3,+4, \ldots
$$
Markov Processes
- A discrete-time random process $X_{n}$ is said to be a Markov process if the process future and past are conditionally independent given its present value
Mathematically this can be rephrased in several ways. For example, if the r.v.s $\left{X_{n}: n \geq 1\right}$ are discrete, then the process is Markov iff
$$
p_{X_{n+1} \mid \mathbf{X}^{n}}\left(x_{n+1} \mid x_{n}, \mathbf{x}^{n-1}\right)=p_{X_{n+1} \mid X_{n}}\left(x_{n+1} \mid x_{n}\right)
$$
for every $n$
- IID processes are Markov
The random walk process is Markov. To see this consider
$$
\begin{aligned}
\mathrm{P}\left{X_{n+1}=x_{n+1} \mid \mathbf{X}^{n}=\mathbf{x}^{n}\right} &=\mathrm{P}\left{X_{n}+Z_{n+1}=x_{n+1} \mid \mathbf{X}^{n}=\mathbf{x}^{n}\right} \
&=\mathrm{P}\left{X_{n}+Z_{n+1}=x_{n+1} \mid X_{n}=x_{n}\right} \
&=\mathrm{P}\left{X_{n+1}=x_{n+1} \mid X_{n}=x_{n}\right}
\end{aligned}
$$
经济代写请认准UpriviateTA. UpriviateTA为您的留学生涯保驾护航。
更多内容请参阅另外一份Galois代写.
线性代数代写
Math 152 lab
Faculty Teaching the Course:
This course is taught by the below faculty~ you may click on their name to view their website with additional information. Please check the Course Explorer or Enterprise/Self-Service to see what section they will be teaching (teaching schedules vary by semester).
Eric McDermott
Joe Petry
Past Course Syllabi:
The following syllabi are from past semesters and should only be used as a guide for the information covered in the course and general structure of the course. The instructors have the right to change the course for upcoming semesters ~ please refer to the syllabus they distribute the first day of class.
Econ 103 Macroeconomic Principles Eric McDermott Past Syllabus
View the Course Explorer:
See when the course is offered in upcoming semesters and all the section details:
https://courses.illinois.edu/schedule/terms/ECON/103
Section Information:
- AL1 or AL2 Lecture MUST be paired with an AQ Discussion (Quiz) Section, and they must be added at the same time when registering. This is in-class for both the lecture twice per week and the discussion.
- No overrides will be provided (sections are not restricted; we do not provide overrides for full sections).
Restrictions will be lifted after priority registration so all students may register for any open seats. No overrides will be provided (students should contact the LLC if they are having problems with registration to ensure they have the student attribute).
