With a continuous distribution, the probability of picking any one number at all is zero, regardless of it being rational or irrational. So it has to be discrete. Now the tough question is, how do you specify a discrete distribution over a nonatomic domain (a domain without a "smallest unit")? May not be possible. E.g. the pseudorandom generators that are programmed into statistical software are defined on rationals only (for example, they will never return the number "pi," just a rational approximation to an arbitrary yet finite number of digits).mruncleramos said:Take the Interval [0,1] over the reals. Randomnly choosing a number, what is the probability that you will get an irrational number? A rational one?
Can you give an example?Hurkyl said:You don't need a discrete distribution to ask about the probability of picking a rational number vs a nonrational number. You don't even need it to be discrete for one (or both) of those probabilities to be nonzero.
Incorrect. I get a number with probability 1. :tongue:Sure, you can always get a set of numbers, but you'll never get a number from a continuous distribution (with positive probability, that is).
That's what I realized yesterday, after Hurkyl's last post. Thanks, all.HallsofIvy said:EnumElish: there is a difference between "getting a specific number" and "getting a number". If I "pick a number at random" then I have to get a number!
I think what you mean is "we can remove a finite, and at most a countably infinite, number of points and still get the same sum."matt grime said:Your idea of throwing away ione point at a time shows that we can remove any finite number of points. it doesn't then follow that we can remove an infinite number of points. We see that we can always remove a countable number of points, but not an uncountable number (we could remove all the points and that would certainly affect the integral).
As there are infinitely more irrational numbers than rational in this interval, the probabilily of getting a rational must be 0 while irrationals are obtained with prob = 1.mruncleramos said:Take the Interval [0,1] over the reals. Randomnly choosing a number, what is the probability that you will get an irrational number? A rational one?