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Unimodality of Probability Measures (Hardcover, 1997 ed.)
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Unimodality of Probability Measures (Hardcover, 1997 ed.)
Series: Mathematics and Its Applications, 382
Expected to ship within 10 - 15 working days
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Labor omnia vincit improbus. VIRGIL, Georgica I, 144-145. In the
first part of his Theoria combinationis observationum erroribus
min- imis obnoxiae, published in 1821, Carl Friedrich Gauss [Gau80,
p.10] deduces a Chebyshev-type inequality for a probability density
function, when it only has the property that its value always
decreases, or at least does l not increase, if the absolute value
of x increases . One may therefore conjecture that Gauss is one of
the first scientists to use the property of 'single-humpedness' of
a probability density function in a meaningful probabilistic
context. More than seventy years later, zoologist W.F.R. Weldon was
faced with 'double- humpedness'. Indeed, discussing peculiarities
of a population of Naples crabs, possi- bly connected to natural
selection, he writes to Karl Pearson (E.S. Pearson [Pea78, p.328]):
Out of the mouths of babes and sucklings hath He perfected praise!
In the last few evenings I have wrestled with a double humped
curve, and have overthrown it. Enclosed is the diagram...If you
scoff at this, I shall never forgive you. Not only did Pearson not
scoff at this bimodal probability density function, he examined it
and succeeded in decomposing it into two 'single-humped curves' in
his first statistical memoir (Pearson [Pea94]).
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