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## Main Question or Discussion Point

Hi All,

Is it possible to express the kurtosis [itex]\kappa[/itex], or the 4th central moment [itex]\mu_4[/itex], of a random variable [itex]X[/itex] in terms of its mean [itex]\mu = E(X)[/itex] and variance [itex]\sigma^2 = Var(X)[/itex] only, without having to particularize to any distribution?

I mean, an expression like [itex]\kappa = f(\mu, \sigma^2)[/itex] or [itex]\mu_4 = g(\mu, \sigma^2)[/itex], valid for any distribution, where [itex]f(\mu, \sigma^2)[/itex] and [itex]g(\mu, \sigma^2)[/itex] are functions of the mean [itex]\mu[/itex] and variance [itex]\sigma^2[/itex].

Thanks in advance!

Chopin

P.S.: Some comments on my attempts.

[itex]\kappa[/itex] is related to [itex]\mu_4[/itex], and [itex]\mu_4 = E(X^4) - 4\mu E(X^3) + 6\mu^2 E(X^2) - 3\mu^4[/itex].

The term [itex]E(X^2)[/itex] can be expressed as [itex]E(X^2) = \mu^2 + \sigma^2[/itex] but I didn't manage to find the way to express [itex]E(X^3)[/itex] and [itex]E(X^4)[/itex] in terms of [itex]\mu[/itex] and [itex]\sigma^2[/itex].

Is it possible to express the kurtosis [itex]\kappa[/itex], or the 4th central moment [itex]\mu_4[/itex], of a random variable [itex]X[/itex] in terms of its mean [itex]\mu = E(X)[/itex] and variance [itex]\sigma^2 = Var(X)[/itex] only, without having to particularize to any distribution?

I mean, an expression like [itex]\kappa = f(\mu, \sigma^2)[/itex] or [itex]\mu_4 = g(\mu, \sigma^2)[/itex], valid for any distribution, where [itex]f(\mu, \sigma^2)[/itex] and [itex]g(\mu, \sigma^2)[/itex] are functions of the mean [itex]\mu[/itex] and variance [itex]\sigma^2[/itex].

Thanks in advance!

Chopin

P.S.: Some comments on my attempts.

[itex]\kappa[/itex] is related to [itex]\mu_4[/itex], and [itex]\mu_4 = E(X^4) - 4\mu E(X^3) + 6\mu^2 E(X^2) - 3\mu^4[/itex].

The term [itex]E(X^2)[/itex] can be expressed as [itex]E(X^2) = \mu^2 + \sigma^2[/itex] but I didn't manage to find the way to express [itex]E(X^3)[/itex] and [itex]E(X^4)[/itex] in terms of [itex]\mu[/itex] and [itex]\sigma^2[/itex].

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