1 Introduction to stochastic processes 1.1 An illustrative class experiment 1.3 Conditional expectation

1.2 Some basic definitions

Definition 1.2.1.

A Stochastic Process (SP) $X$ is a collection of random variables (rvs)

\left\{X(t)\ :\ t\in T\right\}

where $T$ is some index set.

Example 1.2.2.

$T$ is commonly the set of non-negative integers $\{0,\,1,\,2,\,3,\,\ldots\}$ and the process becomes

X_{0},\,X_{1},\,X_{2},\,\ldots.

Our experiment produced such a process. In this case we usually consider $T$ to represent discrete time and write $X_{t}$ in preference to $X(t)$ .

Example 1.2.3.

$T$ is the set of non-negative real numbers, usually considered to be continuous time, and retain the notation $X(t)$ . The number of phone calls since you came onto campus is one such process.

Figure 1.1: Link, Caption: none provided

Definition 1.2.4.

The state space of a stochastic process is the set of possible values of $X(t)$ .

Remark.

This is usually a set which encompasses the values which $X$ may take over the whole time range. In our experiment it would be the non-negative integers even though it is clearly not possible, for example, for $X_{1}$ to be other than 0 or 2. We can consider it to take other values with zero probability.