Pick a random number according to which distribution?
You sound like you want a uniform distribution (i.e., P(x in [a, b]) = b - a / total support of D), but when you have an infinite support, the denominator is infinity - 0 = infinity, so the probability that x is in any finite interval in that set is 0. I've never taken real analysis, so my knowledge here is quite shaky, but I'm not even certain that the resulting probability distribution function is well-defined as a probability distribution function in the first place.
Real analysis, aka, real numbers (and consequently infinite sets) are far weirder than you ever expected them to be.
Yes, you can't define a uniform distribution on the reals at all, there's no way to make it to sum up to 1, which is required. Either it's 0 everywhere and the cumulative probability is 0, or it's a positive constant everywhere and the integral diverges, or it's not uniform.
You sound like you want a uniform distribution (i.e., P(x in [a, b]) = b - a / total support of D), but when you have an infinite support, the denominator is infinity - 0 = infinity, so the probability that x is in any finite interval in that set is 0. I've never taken real analysis, so my knowledge here is quite shaky, but I'm not even certain that the resulting probability distribution function is well-defined as a probability distribution function in the first place.
Real analysis, aka, real numbers (and consequently infinite sets) are far weirder than you ever expected them to be.