# [Probability] question

1. Mar 29, 2010

### zetafunction

let be a Random variable X with a probability defined by a positive measure

$$P(X\lex ) =\int_{-\infty}^{x}dt\mu(t)$$

so $$\mu (x) \ge 0$$ for every Real number $$\mu (x) =\mu(-x)$$ (even measure) and

$$1 =\int_{-\infty}^{\infty}dt\mu(t)$$ (normalization)

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

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

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