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(X). Its components are the expectations of the marginal distributions of the 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 ⁿ to that defines a random variable f(X). We may generalize [3.5] and [3.6] to calculate the mean of f(X). If X is discrete with PF ϕ, we have

[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_j be components of a random vector. We define their (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 X as all its joint moments for which k + l = n. We define its n^th central moments as all its joint central moments for which 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_j) or Σ_i_,j. By definition, covariance is symmetric, with Σ_i,j = Σ_j,i. Also, the covariance of any component X_i with itself is that component’s variance:

[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_j) or ρ_i_,j, of the i^th and j^th components of a random vector X is defined as

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