Stat 5101 Lecture Notes - School of Statistics

1.3. RANDOM VECTORS 151.3.1 Discrete Random VectorsA real-valued function f on a countable subset S of R n is the probabilitydensity (Lindgren would say p. f.) of a discrete random vector if it satisfies thefollowing two propertiesf(x) ≥ 0, for all x ∈ S (1.20a)∑f(x) = 1(1.20b)x∈SThe corresponding probability measure (“big P ”) is defined byP (A) = ∑ x∈Af(x) (1.20c)for all events A (events being, as usual, subsets of the sample space S).Except for the boldface type, these are exactly the same properties thatcharacterize probability densities and probability measures of a discrete randomscalar. The only difference is that x is really an n-tuple, so f is “really” afunction of several variables, and what looks simple in this notation, may becomplicated in practice. We won’t give an example here, but will wait and makethe point in the context of continuous random vectors.1.3.2 Continuous Random VectorsSimilarly, a real-valued function f on a subset S of R n is the probabilitydensity (Lindgren would say p. d. f.) of a continuous random vector if it satisfiesthe following two propertiesf(x) ≥ 0, for all x ∈ S (1.21a)∫f(x) dx = 1(1.21b)The corresponding probability measure is defined by∫P (A) = f(x)dxSA(1.21c)for all events A (events being, as usual, subsets of the sample space S).Again, except for the boldface type, these are exactly the same propertiesthat characterize probability densities and probability measures of a continuousrandom scalar. Also note that the similarity between the discrete and continuouscases, the only difference being summation in one and integration in the other.To pick up our point about the notation hiding rather tricky issues, we goback to the fact that f is “really” a function of several random variables, sothe integrals in (1.21b) and (1.21c) are “really” multiple (or iterated) integrals.Thus (1.21c) could perhaps be written more clearly as∫∫ ∫P (A) = ··· f(x 1 ,x 2 ,...,x n )dx 1 dx 2 ··· dx nA