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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!

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  • $\begingroup$ Normal random variable is unbounded with finite mean and variance. Cauchy random variable is unbounded with infinite mean and variance. $\endgroup$ – Kavi Rama Murthy Oct 31 '18 at 5:45
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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.

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