A289090 Decimal expansion of (E(|x|^3))^(1/3), with x being a normally distributed random variable.

1, 1, 6, 8, 5, 7, 5, 2, 5, 4, 9, 6, 2, 4, 6, 5, 5, 4, 8, 6, 7, 0, 4, 7, 6, 0, 1, 1, 0, 9, 7, 6, 8, 5, 2, 7, 1, 0, 6, 0, 5, 2, 4, 0, 4, 8, 1, 6, 7, 9, 0, 7, 9, 7, 2, 3, 8, 3, 5, 1, 6, 2, 8, 7, 4, 2, 3, 4, 1, 5, 2, 9, 3, 8, 8, 8, 7, 8, 5, 4, 6, 5, 2, 7, 8, 7, 1, 4, 2, 3, 4, 2, 8, 3, 8, 3, 4, 9, 3, 9, 6, 7, 3, 1, 3

Offset: 1

Author

Stanislav Sykora, Jul 26 2017

Keywords

• nonn
• cons

Comments

The p-th root r(p) of the expected value E(|x|^p) for various distributions appears, for example, in chemical physics, where some interactions depend on high powers of interatomic distances.

When x is distributed normally with zero mean and standard deviation 1, r(p) evaluates to r(p) = ((p-1)!!*w(p))^(1/p), where w(p) = 1 for even p and sqrt(2/Pi) for odd p. Note that, by definition, r(2) = 1 and r(1) = w(1) = A076668.

The present constant is a = r(3).

Examples

			1.16857525496246554867047601109768527106052404816790797238351628742...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..1000
• Wikipedia, Normal distribution

Crossrefs

Cf. A060294, A076668 (p=1), A011002 (p=4), A289091 (p=5), A011350 (p=6).

Programs

Mathematica

ExpectedValue[Abs[#]^3&, NormalDistribution[0, 1]]^(1/3) // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Jul 28 2017 *)

PARI

\\ General code, for any p > 0:
r(p) = (sqrt(2/Pi)^(p%2)*prod(k=0,(p-2)\2,p-1-2*k))^(1/p);
a = r(3) \\ Present instance

Formula

Equals (2!!*sqrt(2/Pi))^(1/3) = (2*A076668)^(1/3).