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 Xi:
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
If X is continuous with PDF ϕ, this becomes
3.4.2 Joint moments
Joint moments generalize moments. Let Xi and Xj be components of a random vector. We define their (k, l) joint moment as
We define their (k, l) joint central moment as
We define the nth moments of a random vector X as all its joint moments for which k + l = n. We define its nth central moments as all its joint central moments for which k + l = n.
We are primarily interested in the (1,1) joint central moment, which we call covariance. For the ith and jth components of X, we denote covariance cov(Xi, Xj) or Σi,j. By definition, covariance is symmetric, with Σi,j = Σj,i. Also, the covariance of any component Xi with itself is that component’s variance:
We summarize all the covariances of a random vector X with a covariance matrix:
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.
The correlation, cor(Xi,Xj) or ρi,j, of the ith and jth components of a random vector X is defined as
By construction, a correlation is always a number between –1 and 1. Correlation inherits the symmetry property of covariance: ρi,j = ρj,i. From [3.20] and [3.22], ρi,i = 1, which indicates that a random variable co-varies perfectly with itself. If Xi and Xj 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:
Use [3.17] to prove that, if the components X1 and X2 of a two-dimensional random vector X are independent, then
Consider the two-dimensional discrete random vector Q with PF
Give an example of a two-dimensional random vector whose components have 0 covariance but are not independent.