Originally Posted by

**RanDom** Studying analysis for the first time has made me realise I don't understand what discrete and continuous are as well as I thought I did! I've arrived at what seems to be a couple of paradoxes to me... I'd be be grateful if someone could help me resolve them

A discrete random variable is defined as one that can take a finite **or countably infinite** number of values... Consider choosing a natural number at random... the probability of choosing any particular one of them is zero since there are an infinite number of them and they are all equally likely... But then the expected value is zero using the discrete RV formula (since the natural numbers are countably infinite)... ? This doesn't seem right!

Also, the irrational numbers are uncountable... so I suppose that choosing one at random must be a continuous random variable... but this is very counter-intuitive since there is a rational number between any two irrationals...