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?
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.
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).
Okay, you may be right. It may take a little more to sink in.Hurkyl said:Incorrect. I get a number with probability 1. :tongue:
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).
Do you mean "sets containing an uncountable number of points"?Hurkyl said:there are some uncountable sets of points
You juggled finite vs. infinite, and then countable vs. uncountable. I was just paraphrasing you in a way that (I thought) is marginally more organized.matt grime said:and what was the point of that post?
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?