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MathGroup Archive 2005

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Mean of skew-normal distribution

  • To: mathgroup at smc.vnet.net
  • Subject: [mg61877] Mean of skew-normal distribution
  • From: "Valeri Astanoff" <astanoff at yahoo.fr>
  • Date: Thu, 3 Nov 2005 04:58:46 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Dear group,

I want to prove with help of mathematica [5.1] that this integral :


Integrate[x*(1+Erf[(\[Lambda]*(x-\[Mu]))/
	(Sqrt[2]*\[Sigma])])/(E^((x-\[Mu])^2/(2*\[Sigma]^2))*
	(Sqrt[2*Pi]*\[Sigma])),{x,-Infinity,Infinity}]

 - which is the mean of a so-called skew-normal distribution -
is equal to :

\[Mu] + (Sqrt[2]*\[Lambda]*\[Sigma])/Sqrt[Pi(1 + \[Lambda]^2)]


Series expanding with lambda near 0 shows
that it is true at any order, but of course
that's not a proof, and I shall be grateful
to the good Samaritan that will help me. 


Valeri Astanoff


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