推 llrabel :Good! 01/16 21:17
※ 引述《llrabel (不屈不撓)》之銘言：
: ※ 引述《physmd (smd)》之銘言：
: : 我變分法是很多年前學的，而且只懂皮毛......
: : 底下這個問題不好意思我用英文輸入啦 :P
: : 有請版友指點，多謝～～
: : Consider real valued functions in the closed domain [0, 1]
: : Find f(x) that minimizes: Integrate{ g(x)^2 / f(x) },
: : with respect to a given g(x), integrate over x from 0 to 1,
: : subject to the constraints
: : (1) f > 0 for all x in [0, 1]
: : (2) |f| = 1, that is, Integrate{ f(x) } over x = 0 to 1 is unity.
: : Suppose everything you need to know about g(x) is completely specified.
: : (the derivative or anti-derivative of g(x) etc)
: : In general g(x) can be just some real valued integrable function (that doesn't
: : even have to be piece-wise continuous), but if needed we can focus on the solution
: : f(x) for some smooth g(x).
: : p.s.
: : Some of you might notice that f is a probability density.
: : In fact, this question is simplified from a question regarding minimizing
: : the variance of some stuff.
其實不用變分也行，給一個一行證明：
(\int |g|)^2 \leq \int |g|^2/f \int f =\int |g|^2/f
by Cauchy's ineq.
"=" iff f = c|g|^2/f.
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◆ From: 140.247.0.116
※ 編輯: JASS0213 來自: 140.247.0.116 (01/16 19:45)