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The standard normal (Gaussian) distribution has zero-mean and unit-variance.

I wonder whether there is a zero-mean, unit-variance, and non-normal (non-Gaussian) distribution or not?

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    $\begingroup$ @Sullysaurus, it depends on the random variable's distribution. For instance, what happens if you try to do this to Cauchy variable? $\endgroup$
    – Aksakal
    Commented Jul 22, 2016 at 13:53
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    $\begingroup$ Unlimited number of them, e.g. continous uniform from $-(\sqrt 12) /2$ to $(\sqrt 12) /2$ ... $\endgroup$
    – Tim
    Commented Jul 22, 2016 at 13:53
  • $\begingroup$ @Aksakal Yeah, that's a good point. I'm way off base here. $\endgroup$
    – Quasar
    Commented Jul 22, 2016 at 14:03
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    $\begingroup$ Search our site for standardize: this is a procedure that will turn any distribution with finite second moment into one with zero mean and unit variance. $\endgroup$
    – whuber
    Commented Jul 22, 2016 at 14:03
  • $\begingroup$ You can scale the laplace or the logistic distributions to unit variance and the moments will be matched. $\endgroup$
    – JohnK
    Commented Jul 22, 2016 at 14:07

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There's probably infinite number of distributions in location-scale family, e.g. Student t (location scale). You can convert many distributions into zero mean and unit variance by using this technique.

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    $\begingroup$ Are you really trying to imply the fourth moment of any Gaussian distribution is zero? If not, then please elaborate on what you mean by "only two first moments equal to 0 and 1." $\endgroup$
    – whuber
    Commented Jul 22, 2016 at 14:02
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    $\begingroup$ @whuber, that's Fri morning mistake, long week. $\endgroup$
    – Aksakal
    Commented Jul 22, 2016 at 14:04

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