# Lesson 47 Random Processes

## Motivation

A signal is a function of time, usually symbolized \(x(t)\) (or \(x[n]\), if the
signal is discrete).
In a *noisy* signal, the exact value of the signal is
random. Therefore, we will model noisy signals as a
random function \(X(t)\), where at each time \(t\),
\(X(t)\) is a random variable. These “noisy signals” are
formally called **random processes** or **stochastic processes**.

## Theory

**Definition 47.1 (Random Process) **A **random process** is a collection of random variables \(\{ X_t \}\)
indexed by time. Each realization of the process is a function of \(t\).
For every fixed time \(t\), \(X_t\) is a random variable.

**continuous-time**or

**discrete-time**, depending on whether time is continuous or discrete. We typically notate continuous-time random processes as \(\{ X(t) \}\) and discrete-time processes as \(\{ X[n] \}\).

We have actually encountered several random processes already.

**Example 47.1 (Poisson Process) **The Poisson process, introduced in Lesson 17, is
a continuous-time random process.

Define \(N(t)\) to be the *number of arrivals up to time \(t\)*.
Then, \(\{ N(t); t \geq 0 \}\) is a continuous-time random process.

We can now restate the defining properties of a Poisson process (Definition 17.1) using \(\{ N(t) \}\).

- \(N(0) = 0\).
- \(N(t_1) - N(t_0)\), the number of arrivals on the interval \((t_0, t_1)\), follows a Poisson distribution with \(\mu = \lambda (t_1 - t_0)\)
**Independent increments:**The number of arrivals on non-overlapping intervals are independent.

Shown below are 30 realizations of the Poisson process. At any time \(t\), the value of the process is a discrete random variable that takes on the values 0, 1, 2, ….

**Example 47.2 (White Noise) **In several lessons (for example, Lesson 32 and 46), we have
examined sequences of independent and identically distributed (i.i.d.) random variables.
A sequence of independent and identically distributed random variables
\(.., Z[-2], Z[-1], Z[0], Z[1], Z[2], ...\) is called **white noise**.
White noise is an example of a discrete-time process.

Shown below are 30 realizations of the white noise process. Notice how
the distribution of \(Z[n]\) looks similar for every \(n\). This is because
we constructed the process by simulating an *independent* standard normal
random variable at every time \(n\).

**Example 47.3 (Random Walk) **In Lesson 31, we studied the random walk. More precisely,
we studied a special case called the **simple random walk**.

In general, a **(general) random walk** \(\{ X[n]; n \geq 0 \}\) is a discrete-time process, defined by
\[\begin{align*}
X[0] &= 0 \\
X[n] &= X[n-1] + Z[n] & n \geq 1,
\end{align*}\]
where \(\{ Z[n] \}\) is a white noise process. In other words, each step is a independent and
random draw from the same distribution.

Let’s work out an explicit formula for \(X[n]\) in terms of \(Z[1], Z[2], ...\). \[\begin{align*} X[0] &= 0 \\ X[1] &= \underbrace{X[0]}_0 + Z[1] = Z[1] \\ X[2] &= \underbrace{X[1]}_{Z[1]} + Z[2] = Z[1] + Z[2] \\ X[3] &= \underbrace{X[2]}_{Z[1] + Z[2]} + Z[3] = Z[1] + Z[2] + Z[3] \\ & \vdots \\ X[n] &= Z[1] + Z[2] + \ldots + Z[n]. \end{align*}\]

In a simple random walk, the steps are i.i.d. random variables with p.m.f. \[ \begin{array}{r|cc} z & -1 & 1 \\ \hline f(z) & 0.5 & 0.5 \end{array}. \] See Lesson 31 for pictures of a simple random walk.

Below, we show one realization of a random walk, where the steps \(Z[n]\) are i.i.d. standard normal random variables (i.e., the process considered in Example 47.2).Now, we show 30 realizations of the same random walk process. Notice how the distribution of \(X[n]\) is different for each \(n\). In the Essential Practice below, you will work out the distribution of each \(X[n]\).

## Essential Practice

Radioactive particles hit a Geiger counter according to a Poisson process at a rate of \(\lambda=0.8\) particles per second. Let \(\{ N(t); t \geq 0 \}\) represent this Poisson process.

- What is the distribution of \(N(1.2)\)? (Hint: Translate this into a statement about the number of arrivals on some interval.) Calculate \(P(N(1.2) > 1)\).
- What is \(P(N(2.0) > N(1.2))\)? (Hint: Translate this into a statement about the number of arrivals on some interval.)

Let \(\{Z[n]\}\) be white noise consisting of i.i.d. \(\text{Exponential}(\lambda=0.5)\) random variables.

- What is \(P(Z[2] > 1.0)\)?
- What is \(P(Z[3] > Z[2])\)?

Let \(\{ X[n] \}\) be a random walk, where the steps are i.i.d. standard normal random variables. What is the distribution of \(X[n]\)? (Your answer should depend on \(n\).) What is \(P(X[100] > 20)\)?

(Hint: What do you know about the sum of independent normal random variables?)