Let $X$ be a mean-zero random variable. It has $E|X|^4<\infty$, but $E|X|^k=\infty$ for $k>4$. Is there any famous distribution in literature satisfying this property of $X$?
Any famous distribution only has finite 4-th absolute moment
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statistics
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0I dont know of any *famous* distribution, but it is easy to construct examples, eg, $f(x)=\frac{1}{|x|^{n+2}+1}$ has $E|x|^n<\infty$ but $E|x|^{n+1}=\infty$. – 2017-01-01
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1An example you might actually encounter could be a Student's $t$-distribution with $5$ degrees of freedom (e.g. working with the results from sampling $6$ i.i.d. normal random variables) which has finite $E|X|^4$, but $E|X|^k=\infty$ for $k \ge 5$ – 2017-01-02