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symmetric random variable

Defines: 
symmetric distribution function
Type of Math Object: 
Definition
Major Section: 
Reference
Groups audience: 

Mathematics Subject Classification

60E99 no label found60A99 no label found

Comments

The remark according to which if a random variable XXX is symmetricPlanetmathPlanetmath, then E[X]EXE[X] exists, does not seem to be correct. For example, if a random variable has a Cauchy–Lorentz distributionPlanetmathPlanetmath, i.e., its probability density function is

f(x)=1πγ[1+(x-x0γ)2],fx1πγ1superscriptxsubscriptx0γ2f(x)=\frac{1}{\pi\gamma[1+(\frac{x-x_{0}}{\gamma})^{2}]},

with location parameter x0=0subscriptx00x_{0}=0 and scale parameter γ>0γ0\gamma>0, then it is symmetric, but its mean is undefined.

I think that the correct statement is that if XXX is symmetric *and* E[X]EXE[X] exists, then E[X]=0EX0E[X]=0.

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