let be a Random variable X with a probability defined by a positive measure(adsbygoogle = window.adsbygoogle || []).push({});

[tex] P(X\lex ) =\int_{-\infty}^{x}dt\mu(t) [/tex]

so [tex] \mu (x) \ge 0 [/tex] for every Real number [tex] \mu (x) =\mu(-x) [/tex] (even measure) and

[tex] 1 =\int_{-\infty}^{\infty}dt\mu(t) [/tex] (normalization)

my question is , is there any way to proof that the Polynomial

[tex] 1+\frac{E(t^{2})}{2!}x^{2}+\frac{E(t^{4})}{4!}x^{4}+\frac{E(t^{6})}{6!}x^{6}+\frac{E(t^{8})}{8!}x^{8}+ ..... =E(exp(tx) [/tex]

has ONLY pure imaginary roots ?? , for example if the measure is '1' defined on (-1,1) then we have [tex]sinh(x)/x [/tex] which have only pure imaginary roots.

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# [Probability] question

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