Random Vectors

Consider the example of the confined aquifer in section 1.1.2. An infinite family $\cal F$ of random variables

$$\{ Y_x, x \in S \}$$ (1.6)

was constructed there, where

$$Y_x(\omega) := h(x, \omega)$$

is the piezometric head at $x$ corresponding to a transmissivity profile $\omega$.

Suppose a number of points $p_1, \ldots, p_n \in S$ are selected for measurement and let the random variables $X_1, \ldots, X_n$ be defined by

$$X_i := Y_{p_i}, \quad i=1, \ldots, n.$$

Then, $X := (X_1, \ldots, X_n)$ constitutes a random vector: it is a vector valued random quantity.

Just as in the scalar case, the probabilities of the components of a random vector $X$ are embodied in its joint distribution function $F_X :$   $^n \rightarrow$   , defined as follows:

$$F_{X_1, \ldots, X_n} (x_1, \ldots, x_n) = P(X_1 \le x_1, \ldots, X_n \le x_n)$$

$$= P((X_1 \le x_1) \cap X_2 \le x_2 \cap \ldots \cap X_n \le x_n)$$

For instance if we throw two dice at the same time the outcome of one does not depend on the outcome of the other. Then we can have: zsh: command not found: G if its density is

$$f_X(x) = {1 \over \sqrt{2 \pi}} e^{-\Vert x\Vert^2 / 2}$$

where $\Vert\cdot\Vert$ denotes the ordinary Euclidean norm in $^n$.

The expectation of a random vector $X$ is defined componentwise:

$$E[X] := (E[X_1], \ldots, E[X_n])^T$$

The covariance matrix of $X$ is defined as:

   Cov$$(X) := E[(X - E[X])(X - E[X])^T]$$

The $(i,j)$-th element of Cov$(X)$ is

$$c_{ij} := E[(X_i - E[X_i])(X_j - E[X_j])^T]$$

and it is referred to as the covariance of $X_i$ and $X_j$.

A simple computation shows that if $X \sim N({\mathbf 0}, I)$, then $X$ has mean ${\mathbf 0} \in$   $^n$ and its covariance matrix is the identity matrix $I \in$   $^{n \times n}$.