1) Is there such thing as an unbounded random variable? Can a variable attain values of infinity (I thought this was only something you could tend towards)? Or would the probability of reaching infinity just be zero?
2) If so, can a random variable have unbounded expectation or variance?
Simple examples are much appreciated, thanks!
Consider random variable defined on set of atural numbers $\mathbb N$ as,
$X:\mathbb N \to \mathbb R$ and $X(n)=n$
Note that above random variable is unbounded and also its expectation is infinity.
If you know concept of measure then random variable is just a measurable function.