| [ QuizWit ] in KIDS 글 쓴 이(By): guest (wiking) 날 짜 (Date): 1998년 5월 9일 토요일 오전 07시 57분 04초 제 목(Title): Re: probability 2 Claim1: Given statement is equal to E[f*g] >= 0 if f, g bounded increasing & conti. while E[f]=E[g]=0. Step 1: If A<=0 in [-inf, a], A>=0 in [a, inf] while integral(A) = 0 and B is positive increasing, then intergral(AB) >= 0 Proof> int[a, inf](AB) >= B(a)int[a,inf](A) = B(a)int[-inf,a](-A) >= int[-inf, a](-BA). Q.E.D Step 2: Define A = p*g, f = fp - fn where fp = f*theta(x-b), fn = -f*theta(b-x) where b is the unique zero of f. Then from Step 1, int(A*fp) >= while int(A*fn) <=0 because fp is positive increasing, fn is positive decreasing. Therefore E[f*g]=int(A*f)>=0. |