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it might be a stupid question but I was discussing with a colleague when the 3rd axiom of probability (sigma additivity) is really needed. I argue that in the case of a discrete distribution, say a single die, the first two axioms are sufficient as this distribution has a finite number of events. Is this right?

And then I am stretching it a bit by arguing that for simple/well behaved continuous distributions, such as the uniform distribution, the first two axioms are sufficient to ensure a proper probability distribution. Is that right?

I always had the idea that the 3rd axiom was rather there to ensure that more cumbersome distributions or convolutions of distributions would still be proper probability distributions (although I don't have an example at hand). Or is axiom 3 much more vital than I understand?

Thanks for any input! Best, Stefan

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    I don't understand what "for simple/well behaved continuous distributions... the first two axioms are sufficient to ensure a proper probability distribution" means. Surely "satisfies sigma-additivity" is a necessary condition for being simple and/or well-behaved.2011-01-04
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    Note that discrete distributions need not be finite. Perhaps your point is that for a finite event space, finite additivity will suffice.2011-01-04

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