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