In article <CfhJ4.10505$[EMAIL PROTECTED]>,
Aaron & Katya <[EMAIL PROTECTED]> wrote:
>If you distribute your mass 1/2 at 0 and 1/2 at 1 the variance is 1/4. (ie
>Var[x]=E[x^2]-E[x]^2=(0^2*1/2+1^2*1/2)-(0*1/2+1*1/2)^2=1/4.
>I believe this is the maximum variance for the interval [0,1]. Other than
>just making an assertion, is there a way to prove that Var[x]<=1/4.
This is well known. If one has candidates m_i for moments
of order 0, 1, ..., 2k for a distribution on [0,1], a
necessary and sufficient condition that they are moments
of something is that the two matrices ||m_{i+j}||.
i, j = 0, ..., k, and ||m_{i+j+1} - m_{i+j+2}||, where
now the bound is k-1, are positive semidefinite. For
k=1, the conditions are that m_2 >= m_1^2 and m_1 >= m_2.
So the variance m_2 - m_1^2 is bounded by m_1 - m_1^2,
attained for Bernoulli trials, and this is maximized
at m_1 = .5.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
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