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

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• 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}

## Math 152 lab

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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

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### Section Information:

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