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A.2 Random Vectors 375
The probability density of X is then found from
f(x1,...,xn) ∂nF(x1,...,xn) .
∂x1 ···∂xn
The random vector X is said to be discrete if there exist real-valued vectors x0, x1,...
and a probability mass function p(xj) P [X xj] such that
∞
p(xj) 1.
j0
The expectation of a function g of a random vector X is defined by
E (g(X)) g(x)dF(x) g(x1,...,xn)dF(x1,...,xn),
where
g(x1,...,xn)dF(x1,...,xn)
⎧
⎪⎨
··· g(x1,...,xn)f (x1,...,xn)dx1 ···dxn, in the continuous case,
⎪⎩ ···
g(xj1,...,xjn)p(xj1,...,xjn), in the discrete case,
j1
jn
and g is any function such that E|g(X)| < ∞.
The random variables X1,...,Xn are said to be independent if
i.e.,
P [X1 ≤ x1,...,Xn ≤ xn] P [X1 ≤ x1] ···P [Xn ≤ xn],
F(x1,...,xn) FX1(x1) ···FXn(xn)
for all real numbers x1,...,xn. In the continuous and discrete cases, independence
is equivalent to the factorization of the joint density function or probability mass
function into the product of the respective marginal densities or mass functions, i.e.,
or
f(x1,...,xn) fX1 (x1) ···fXn (xn) (A.2.2)
p(x1,...,xn) pX1 (x1) ···pXn (xn). (A.2.3)
For two random vectors X (X1,...,Xn) ′ and Y (Y1,...,Ym) ′ with joint
density function fX,Y, the conditional density of Y given X x is
⎧
⎨ fX,Y(x, y)
, if fX(x) >0,
fY|X(y|x) fX(x)
⎩
fY(y), if fX(x) 0.