Stat 5101 Lecture Notes - School of Statistics

74 Stat 5101 (Geyer) Course NotesExample 2.5.6 (The Gamma Distribution Again).The the Gamma distribution is the continuous distribution having densityf(x) =λαΓ(α) xα−1 e −λx , x > 0(Section B.2.3 of Appendix B). For X ∼ Gam(α, λ), we consider here when X pis in L 1 for any real number p, positive or negative. The integral that definesthe expectation isE(X p )=∫ ∞x p λα0Γ(α) xα−1 e −λx dx =λαΓ(α)∫ ∞0x α+p−1 e −λx dxif the integral exists (which is the question we are examining).From Lemma 2.41, the integral over (a, ∞) exists for for any a>0 and anyp positive or negative. The only issue is the possible singularity of the integrandat the origin. There is a singularity if α + p − 1 < 0. Otherwise the integrandis bounded and the expectation exists.Since e 0 = 1, the integrand behaves like x α+p−1 at zero and according toLemma 2.43 this is integrable over a neighborhood of zero if and only if α+p−1 >−1, that is, if and only if p>−α.2.5.10 L p SpacesWe start with another consequence of the domination theorem and the methodsfor telling when expectations exist developed in the preceding section.Theorem 2.44. If X is a real-valued random variable and |X − a| p is in L 1for some constant a and some p ≥ 1, then|X − b| q ∈ L 1 ,for any constants b and any q such that 1 ≤ q ≤ p.Proof. First the case q = p. The ratio of the integrands defining the expectationsof |X − a| p and |X − b| p converges, that is∣ |x − b| p f(x) ∣∣∣p|x − a| p f(x) = x − bx − a∣goes to 1 as x goes to plus or minus infinity. Thus both integrals exist, and|X − b| p ∈ L 1 .In the case q