The mean vector of an integrable random vector

X = (X_{1}, \dots, X_{d})

is its coordinatewise expected value:

E X = (E X_{1}, \dots, E X_{d})

. Integrability of every coordinate makes this a finite vector.

Planar projection of an isotropic random vector ( $E R_{⊥} = (π /4) E R, E R_{⊥}^{2} = (2/3) E R^{2}$ )

 0  0

For an isotropic three-dimensional random vector, its direction is uniform on the sphere and independent of its length. Projection onto a fixed plane gives

R_{⊥} = R sin θ

. The solid-angle averages are

E sin θ = (1/2) \int_{0}^{π} sin^{2} θ d θ = π /4

and

E sin^{2} θ = (1/2) \int_{0}^{π} sin^{3} θ d θ = 2/3

. This proves the two formulas whenever the corresponding moments exist. The mean-length factor and root-mean-square factor are different. Both are independent of the radial distribution, so projection preserves a polymer's chain-length scaling exponent.

Joint probability mass function ( $p_{X, Y} (x, y) = P (X = x, Y = y)$ )

 0  0

For discrete random variables

X, Y

, their joint probability mass function assigns a probability to each possible ordered pair. Its entries are nonnegative and sum to one. Marginalization gives the probability mass functions

p_{X} (x) = \sum_{y} p_{X, Y} (x, y)

and

p_{Y} (y) = \sum_{x} p_{X, Y} (x, y)

. In a table with rows indexed by

x

and columns by

y

, these are the row and column totals. The joint probability distribution records how the coordinates occur together, while the marginal distributions describe each separately.