# Example where higher moments are infinite

1. Oct 25, 2009

### 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,
$$\mathbb{E}| \log x |< \infty$$

2. Oct 25, 2009

### mathman

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.

3. Oct 26, 2009

### bpet

Try the Pareto distribution.

4. Oct 26, 2009

### St41n

Thank you very much. It makes sense