Editing Hermite polynomials (section)

===Expected values===
If {{mvar|X}} is a [[random variable]] with a [[normal distribution]] with standard deviation 1 and expected value {{mvar|μ}}, then
<math display="block">\operatorname{\mathbb E}\left[\operatorname{He}_n(X)\right] = \mu^n.</math>

The moments of the standard normal (with expected value zero) may be read off directly from the relation for even indices:
<math display="block">\operatorname{\mathbb E}\left[X^{2n}\right] = (-1)^n \operatorname{He}_{2n}(0) = (2n-1)!!,</math>
where {{math|(2''n'' &minus; 1)!!}} is the [[double factorial]]. Note that the above expression is a special case of the representation of the probabilist's Hermite polynomials as moments:
<math display="block">\operatorname{He}_n(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty (x + iy)^n e^{-\frac{y^2}{2}} \,dy.</math>