2 ms·

A normal distribution is infinite so there is always a (small) nonzero expected value for any real number, so technically, even for "mean" much larger than zero

A normal distribution is infinite so there is always a (small) nonzero expected value for any real number, so technically, even for "mean" much larger than zero. However, GP is accidentally reversing the causality, which is incorrect: We say that something follows a normal distribution if the samples we observe fit, e.g. shoe sizes or height, which are clearly non-negative. It doesn't mean that any possible distribution value must occur in the original samples for the population to be normally distributed.

zmgsabst 2y ago
I think the OPs point is that it’s bounded below at a point not-distant from the mean, and hence you have a one-tailed and not two-tailed distribution. That non-trivially changes the distribution. And then commented why: you have a finite amount you can do a task faster; but an infinite amount you can do it slower. That is in contrast to heights or shoe sizes, which are (effectively) bounded on both sides while having the bounds distant from the mean.
- mrks_hy 2y ago
  That is just being pedantic. In reality, there are no observable tasks that take infinite time. And the actual statistical process is probably unknowable anyway, we are very likely dealing with a sum of random variables, hence the Central Limit Theorem applies anyway.
dxdm 2y ago
Thanks for explaining! This stretching to infinity is interesting, because it assigns a probability to outcomes even if they cannot happen, for example where observed values cannot go negative like here. That shows the difference between the "model" and the reality we try to fit it on; and that the model is a tool that just needs to fit well enough to be useful.
- pdpi 2y ago
  Yeah, it’s very much a case of “the map is not the territory”. Modelling such processes as normally distributed is objectively wrong, but it’s also not wrong enough to matter. We could, and arguably should, model such processes as truncated normals instead, but normals are just so damn convenient to work with.
  - baq 2y ago
    You say 'convenience' and I don't know if it's a strong enough word - the gap in difficulty between working with normals and anything else is like in the draw the rest of the owl meme - it's so wide nobody ever bothers.

zmgsabst 2y ago

I think the OPs point is that it’s bounded below at a point not-distant from the mean, and hence you have a one-tailed and not two-tailed distribution. That non-trivially changes the distribution. And then commented why: you have a finite amount you can do a task faster; but an infinite amount you can do it slower. That is in contrast to heights or shoe sizes, which are (effectively) bounded on both sides while having the bounds distant from the mean.

mrks_hy 2y ago
That is just being pedantic. In reality, there are no observable tasks that take infinite time. And the actual statistical process is probably unknowable anyway, we are very likely dealing with a sum of random variables, hence the Central Limit Theorem applies anyway.

dxdm 2y ago

Thanks for explaining! This stretching to infinity is interesting, because it assigns a probability to outcomes even if they cannot happen, for example where observed values cannot go negative like here. That shows the difference between the "model" and the reality we try to fit it on; and that the model is a tool that just needs to fit well enough to be useful.

pdpi 2y ago
Yeah, it’s very much a case of “the map is not the territory”. Modelling such processes as normally distributed is objectively wrong, but it’s also not wrong enough to matter. We could, and arguably should, model such processes as truncated normals instead, but normals are just so damn convenient to work with.
- baq 2y ago
  You say 'convenience' and I don't know if it's a strong enough word - the gap in difficulty between working with normals and anything else is like in the draw the rest of the owl meme - it's so wide nobody ever bothers.