random variables problem

[thenumber42]

Hello i have a few questions in probability i can't make heads or tails of.
hints or answer much appreciated

for question 1 assume E[x*y]=E[x]*E[y]
question 1:
are there dependent random variables x,y such that E[x]!= 0 and E[Y]!= 0
question 2:
are there random variables such that E[x] and E[y] exist but E[x*y]
doesn't?

question 3:
are there random variables x,y such that E[x*y] exist but either E[x] or E[y] doesn't exist?



my approach to this have been exploring families of distributations as the 5 ones we saw in class don't really give a result

(uniform , binomial , geometric , negative-binomial , hyper-geometric)

[BGM]

1) Take
Then

2) http://en.wikipedia.org/wiki/Inverse-gamma_distribution
Since we just require a random variable only have the first moment well defined,
we may take

3) Take

[Dason]

For question 2 Let X=Y and X be distributed as a t-distribution with degrees of freedom in (1, 2] could work as well.