# 3.4 Parameters of Random Vectors

The **expectation** of an *n*-dimensional random vector ** X** is a vector which we denote either

**μ**or

*E*(

**). Its components are the expectations of the marginal distributions of the**

*X**X*:

_{i}[3.14]

###### 3.4.1 Expectation of a function of a random vector

Let ** X** be an

*n*-dimensional random vector and

*f*a function from

*to that defines a random variable*

^{n}*f*(

**). We may generalize [3.5] and [3.6] to calculate the mean of**

*X**f*(

**). If**

*X***is discrete with PF ϕ, we have**

*X*[3.15]

If ** X** is continuous with PDF ϕ, this becomes

[3.16]

[3.17]

###### 3.4.2 Joint moments

Joint moments generalize moments. Let *X _{i}* and

*X*be components of a random vector. We define their (

_{j}*k*,

*l*)

**joint moment**as

[3.18]

We define their (*k*, *l*) **joint central moment** as

[3.19]

We define the ** n^{th} moments** of a random vector

**as all its joint moments for which**

*X**k*+

*l*=

*n*. We define its

**as all its joint central moments for which**

*n*^{th}central moments*k*+

*l*=

*n*.

###### 3.4.3 Covariance

We are primarily interested in the (1,1) joint central moment, which we call **covariance**. For the *i*^{th} and *j*^{th} components of ** X**, we denote covariance

*cov*(

*X*,

_{i}*X*) or Σ

_{j}

_{i}_{,j}. By definition, covariance is symmetric, with Σ

*= Σ*

_{i,j}*. Also, the covariance of any component*

_{j,i}*X*with itself is that component’s variance:

_{i}[3.20]

We summarize all the covariances of a random vector *X** *with a **covariance matrix**:

[3.21]

By the symmetry property of covariances, the covariance matrix is symmetric.

Intuitively, covariance is a metric of the tendency of two components of a random vector to vary together, or co-vary. The magnitude of a covariance depends upon the standard deviations of the two components. To obtain a more direct metric of how two components co-vary, we scale covariance to obtain **correlation**.

###### 3.4.4 Correlation

The **correlation**, *cor*(*X _{i}*,

*X*) or ρ

_{j}

_{i}_{,j}, of the

*i*

^{th}and

*j*

^{th}components of a random vector

**is defined as**

*X*[3.22]

By construction, a correlation is always a number between –1 and 1. Correlation inherits the symmetry property of covariance: ρ_{i}_{,j} = ρ_{j}_{,i}. From [3.20] and [3.22], ρ_{i}_{,i} = 1, which indicates that a random variable co-varies perfectly with itself. If *X _{i}* and

*X*are independent, their correlation is 0. The converse is not true. As with covariances, we can summarize all the correlations of a random vector

_{j}

*X**with a symmetric*

**correlation matrix**:

[3.23]

###### Exercises

Use [3.17] to prove that, if the components *X*_{1} and *X*_{2} of a two-dimensional random vector ** X **are independent, then

[3.24]

Consider the two-dimensional discrete random vector ** Q **with PF

[3.25]

Calculate ρ_{1,2}.

Solution

Give an example of a two-dimensional random vector whose components have 0 covariance but are not independent.

Solution