# 3.2 Prerequisites

We assume familiarity with basic notation and concepts from probability. If *E* is an event, we denote its probability *Pr*(*E*). You should be familiar with **random variables** and **random vectors**. A random vector ** X** can be thought of as an

*n*-dimensional vector of random variables

*X*all defined on the same sample space. When we present general definitions or results for random vectors, these also apply to random variables.

_{i}It is important to distinguish between a random vector ** X** and a realization of that random vector, which we may denote

**. The**

*x***realization**is an element of the range of the random vector.

You should be familiar with **discrete** and **continuous** distributions for random vectors. You should be comfortable working with **probability functions** (PFs), **probability density functions** (PDFs), and **cumulative distribution functions** (CDFs). You should be familiar with **joint distributions**, **conditional distributions**, and **marginal distributions**.

We may think of random vectors as being “equivalent” in several senses. We distinguish between two of these. Random vectors ** X** and

**are**

*Y***equal**, denoted

**=**

*X***, if they both take on the same value with probability 1. If**

*Y***and**

*X***simply have the same probability distribution, we denote this relationship**

*Y***~**

*X***. We also use the symbol ~ to indicate what a random variable represents, say:**

*Y**X*~ tomorrow’s 3-month USD Libor rate.

You should know what it means for two or more components of a random vector ** X** to be

**independent**. In particular, if

*n*components

*X*are independent, their joint CDF and marginal CDFs satisfy:

_{i}[3.1]

for all* x*_{1}, *x*_{2}, … ,* x _{n}* ∈ . Similarly, their joint PDF and marginal PDFs satisfy:

[3.2]

for all *x*_{1}, *x*_{2}, … ,* x _{n}* ∈ .1