Stat 5101 Lecture Notes - School of Statistics

1.2. CHANGE OF VARIABLES 9Thus even constant random variables have probability distributions. Theyare rather trivial, all the probabilities being either zero or one, but they areprobability models that satisfy the axioms.Thus in probability theory we treat nonrandomness as a special case ofrandomness. There is nothing uncertain or indeterminate about a constantrandom variable. When Y is defined as in the example, we always know Y =g(X) = c, regardless of what happens to X. Whether one regards this asmathematical pedantry or a philosophically interesting issue is a matter of taste.1.2.2 Discrete Random VariablesFor discrete random variables, probability measures are defined by sumsP (A) = ∑ x∈Af(x) (1.13)where f is the density for the model (Lindgren would say p. f.)Note also that for discrete probability models, not only is there (1.13) givingthe measure in terms of the density, but alsof(x) =P({x}). (1.14)giving the density in terms of the measure, derived by taking the case A = {x}in (1.13). This looks a little odd because x is a point in the sample space, anda point is not a set, hence not an event, the analogous event is the set {x}containing the single point x.Thus our job in applying the change of variable theorem to discrete probabilitymodels is much simpler than the general case. We only need to considersets A in the statement of the theorem that are one-point sets. This gives thefollowing theorem.Theorem 1.2. If X is a discrete random variable with density f X and samplespace S, and Y = g(X), then Y is a discrete random variable with density f Ydefined byf Y (y) =P X (B)=x∈Bf ∑ X (x),whereB = { x ∈ S : y = g(x) }.Those who don’t mind complicated notation plug the definition of B intothe definition of f Y obtainingf Y (y) =∑ x∈Sy=g(x)f X (x).In words, this says that to obtain the density of a discrete random variable Y ,one sums the probabilities of all the points x such that y = g(x) for each y.Even with the simplification, this theorem is still a bit too abstract andcomplicated for general use. Let’s consider some special cases.