Stat 5101 Lecture Notes - School of Statistics

98 Stat 5101 (Geyer) Course Notes(with integration replaced by summation if the probability model is discrete).In order for c to be a positive number, the integral (or sum in the discretecase) must exist and be nonzero. This gives us two conditions on unnormalizeddensities. A real-valued function h(x) isanunnormalized density provided thefollowing two conditions hold.• It is nonnegative: h(x) ≥ 0, for all x.• It is integrable in the continuous case or summable in the discrete caseand the integral or sum is nonzero.Thenf(x) = 1 c h(x)is a normalized probability density, where c is given by (3.17) in the continuouscase and by (3.17) with the integral replaced by a sum in the discrete case.Example 3.4.2.Consider the functionh(x) =x α−1 e −x , x > 0,where α>0. How do we normalize it to make a probability density?The normalizing constant isc =∫ ∞0x α−1 e −x dx =Γ(α)by (B.2) in Appendix B. Thus we obtain a gamma distribution densityf(x) = 1Γ(α) xα−1 e −x .So what’s the big deal? We already knew that! Is “normalization” just afancy name for something trivial? Well, yes and no. You can form your ownopinion, but not until the end of Section 3.4.3.4.3 RenormalizationWe start with a sloganConditional probability is renormalization.What this means will become apparent presently.First, f(y | x) is just an ordinary probability density when considered as afunction of y for fixed x. We maintain this view, y is the variable and x is fixed,throughout this subsection.Second, since x is fixed, the denominator inf(y | x) =f(x, y)f X (x)(3.18)