Random vectors
Section of Covariance and the multivariate normal, part of Statistical learning
Random vectors
| Type | Section |
| Taught in | Covariance and the multivariate normal |
| Part | Statistical learning |
Write \mathbf{X}_i for the random measurement vector on unit i, and \mathbf{x}_i for its observed value. The index i=1,\ldots,n identifies the unit. The index j=1,\ldots,p identifies a coordinate within that unit. The usual sampling assumption is that \mathbf{X}_1,\ldots,\mathbf{X}_n are independent and identically distributed. That assumption says nothing about independence of the coordinates inside one vector. Unless stated otherwise the coordinates are continuous, and \mathbf{X}=(X_1,\ldots,X_p)^\top is a generic observation. One row of the data matrix is one flower. The four iris measurements are four coordinates of that flower.
Definition 1 (Joint distribution and density) The joint distribution function of \mathbf{X} is
F(\mathbf{x})=\mathbb{P}(\mathbf{X}\le\mathbf{x})=\mathbb{P}(X_1\le x_1,\ldots,X_p\le x_p), \qquad \mathbf{x}\in\mathbb{R}^p.
If F is absolutely continuous, \mathbf{X} has a joint density
f(\mathbf{x})=\frac{\partial^p F(\mathbf{x})}{\partial x_1\cdots\partial x_p}
almost everywhere, and
F(\mathbf{x})=\int_{-\infty}^{x_p}\cdots\int_{-\infty}^{x_1} f(u_1,\ldots,u_p)\,\,\mathrm{d} u_1\cdots\,\mathrm{d} u_p.
Independent coordinates are exactly the case in which the density and the distribution function both factor,
f(\mathbf{x})=\prod_{j=1}^p f_j(x_j), \qquad F(\mathbf{x})=\prod_{j=1}^p F_j(x_j),
with f_j and F_j belonging to X_j alone. A factorisation of this form is independence. In the models below the coordinates are usually dependent, and the point of the vector calculation is to describe that dependence.
A density can be easy to evaluate when its integral over a rectangle has no elementary antiderivative. The correlated multivariate normal is that kind of density: probabilities of rectangles are computed by numerical integration. Probability still comes from integrating over a region. A continuous distribution does not put positive probability on a single point.
Used in
See also
Joint distribution and density · Mean vector and covariance matrix · Correlation · Multivariate normal distribution · Squared Mahalanobis distance · Linear images of a normal vector · Conditional distribution of a multivariate normal · Partial correlation