example where higher moments are infinite

[St41n]

Can someone give me an example where we have \mathbb{E}z=0 , \mathbb{E}z^2=1 (i.e. finite expectations)
BUT,
\mathbb{E}z^4= \infty ?

Also, I cannot think of a case where:
\mathbb{E}x=\infty where x>0
BUT,
\mathbb{E}| \log x |< \infty

Thanks in advance

[mathman]

Can someone give me an example where we have \mathbb{E}z=0 , \mathbb{E}z^2=1 (i.e. finite expectations)
BUT,
\mathbb{E}z^4= \infty ?

Also, I cannot think of a case where:
\mathbb{E}x=\infty where x>0
BUT,
\mathbb{E}| \log x |< \infty

Thanks in advance
For your first question, let the density function f(x)=k/(1+x4).

For the second, f(x)=c/(1+x2) for x>0, f(x)=0 for x<0.

[bpet]

Can someone give me an example where we have \mathbb{E}z=0 , \mathbb{E}z^2=1 (i.e. finite expectations)
BUT,
\mathbb{E}z^4= \infty ?

Also, I cannot think of a case where:
\mathbb{E}x=\infty where x>0
BUT,
\mathbb{E}| \log x |< \infty

Thanks in advance

Try the Pareto distribution.

[St41n]

Thank you very much. It makes sense